diff --git a/src/sage/arith/misc.py b/src/sage/arith/misc.py
index 53d9683279..564b99044f 100644
--- a/src/sage/arith/misc.py
+++ b/src/sage/arith/misc.py
@@ -1458,13 +1458,13 @@ def divisors(n):
sage: K. = QuadraticField(7)
sage: divisors(K.ideal(7))
- [Fractional ideal (1), Fractional ideal (a), Fractional ideal (7)]
+ [Fractional ideal (1), Fractional ideal (-a), Fractional ideal (7)]
sage: divisors(K.ideal(3))
[Fractional ideal (1), Fractional ideal (3),
Fractional ideal (-a + 2), Fractional ideal (-a - 2)]
sage: divisors(K.ideal(35))
- [Fractional ideal (1), Fractional ideal (5), Fractional ideal (a),
- Fractional ideal (7), Fractional ideal (5*a), Fractional ideal (35)]
+ [Fractional ideal (1), Fractional ideal (5), Fractional ideal (-a),
+ Fractional ideal (7), Fractional ideal (-5*a), Fractional ideal (35)]
TESTS::
diff --git a/src/sage/ext_data/pari/simon/ell.gp b/src/sage/ext_data/pari/simon/ell.gp
index 74f0786646..21cff9cbb3 100644
--- a/src/sage/ext_data/pari/simon/ell.gp
+++ b/src/sage/ext_data/pari/simon/ell.gp
@@ -1038,7 +1038,7 @@ if( DEBUGLEVEL_ell >= 1, print(" trivial points on E(K) = ");
KS2gen = KS2gen[1];
for( i = 1, #KS2gen,
KS2gen[i] = nfbasistoalg(bnf, KS2gen[i]));
- KS2gen = concat(Mod(lift(bnf.tufu),bnf.pol),KS2gen);
+ KS2gen = concat(Mod(lift(concat(bnf.tu[2], bnf.fu)),bnf.pol),KS2gen);
if( DEBUGLEVEL_ell >= 2,
print(" #K(b,2)gen = ",#KS2gen);
print(" K(b,2)gen = ",KS2gen));
@@ -1072,7 +1072,7 @@ if( DEBUGLEVEL_ell >= 1,
KS2gen = KS2gen[1];
for( i = 1, #KS2gen,
KS2gen[i] = nfbasistoalg(bnf, KS2gen[i]));
- KS2gen = concat(Mod(lift(bnf.tufu),bnf.pol),KS2gen);
+ KS2gen = concat(Mod(lift(concat(bnf.tu[2], bnf.fu)),bnf.pol),KS2gen);
if( DEBUGLEVEL_ell >= 2,
print(" #K(a^2-4b,2)gen = ",#KS2gen);
print(" K(a^2-4b,2)gen = ",KS2gen));
@@ -1244,11 +1244,11 @@ if( DEBUGLEVEL_ell >= 4, print(" bbbnf.clgp = ",bbbnf.clgp));
SL1 = idealmul(bbbnf,SL0,rnfeltup(rrrnf,bleg));
SL = idealfactor(bbbnf,SL1)[,1]~;
sunL = bnfsunit(bbbnf,SL);
- fondsunL = concat(bbbnf.futu,vector(#sunL[1],i,nfbasistoalg(bbbnf,sunL[1][i])));
+ fondsunL = concat(concat(bbbnf.fu, bbbnf.tu[2]),vector(#sunL[1],i,nfbasistoalg(bbbnf,sunL[1][i])));
normfondsunL = vector(#fondsunL, i, norm(rnfeltabstorel(rrrnf,fondsunL[i])));
SK = idealfactor(bnf,idealnorm(bbbnf,SL1))[,1]~;
sunK = bnfsunit(bnf,SK);
- fondsunK = concat(bnf.futu,vector(#sunK[1],i,nfbasistoalg(bnf,sunK[1][i])));
+ fondsunK = concat(concat(bnf.fu, bnf.tu[2]),vector(#sunK[1],i,nfbasistoalg(bnf,sunK[1][i])));
vecbleg = bnfissunit(bnf,sunK,bleg);
matnorm = matrix(#fondsunK,#normfondsunL,i,j,0);
for( i = 1, #normfondsunL,
@@ -1345,7 +1345,7 @@ if( DEBUGLEVEL_ell >= 4, print("on factorise bb = ",bb));
sun = bnfsunit(bnf,idealfactor(bnf,bb)[,1]~);
fact = lift(bnfissunit(bnf,sun,bb));
if( DEBUGLEVEL_ell >= 4, print("fact = ",fact));
- suni = concat(bnf.futu,vector(#sun[1],i,nfbasistoalg(bnf,sun[1][i])));
+ suni = concat(concat(bnf.fu, bnf.tu[2]),vector(#sun[1],i,nfbasistoalg(bnf,sun[1][i])));
for( i = 1, #suni,
if( (f = fact[i]>>1),
test =0;
@@ -1554,7 +1554,7 @@ if( DEBUGLEVEL_ell >= 3, print(" KS2gen = ",KS2gen[1]));
LS2gen = LS2gen[1];
LS2 = vector(#LS2gen,i,lift(nfbasistoalg(Lrnf,LS2gen[i])));
- LS2 = concat(lift(Lrnf.futu),LS2);
+ LS2 = concat(lift(concat(Lrnf.fu, Lrnf.tu[2])),LS2);
LS2 = subst(LS2,'x,ttheta);
LS2 = LS2*Mod(1,polrel);
@@ -1992,7 +1992,7 @@ if( DEBUGLEVEL_ell >= 2, print(" Algorithm of complete 2-descent"));
KS2gen = KS2gen[1];
for( i = 1, #KS2gen,
KS2gen[i] = nfbasistoalg(bnf, KS2gen[i]));
- KS2gen = concat(Mod(lift(bnf.tufu),bnf.pol),KS2gen);
+ KS2gen = concat(Mod(lift(concat(bnf.tu[2], bnf.fu)),bnf.pol),KS2gen);
if( DEBUGLEVEL_ell >= 2,
print(" #K(S,2)gen = ",#KS2gen);
print(" K(S,2)gen = ",KS2gen)
diff --git a/src/sage/ext_data/pari/simon/ellQ.gp b/src/sage/ext_data/pari/simon/ellQ.gp
index aede9fc941..27cc124372 100644
--- a/src/sage/ext_data/pari/simon/ellQ.gp
+++ b/src/sage/ext_data/pari/simon/ellQ.gp
@@ -1162,7 +1162,7 @@ if( DEBUGLEVEL_ell >= 4, print(" kerval = ",kerval));
LS2gen[j]^kerval[j,i]));
\\ Add the units
- LS2gen = concat(Mod(bnf[8][5],bnf.pol),LS2gen); \\ LS2gen = concat(bnf.fu,LS2gen);
+ LS2gen = concat(bnf.fu,LS2gen); \\ LS2gen = concat(bnf.fu,LS2gen);
\\ Add also the torsion unit if its order is divisible by p.
if( bnf[8][4][1]%p == 0, \\ if( bnf.tu[1]%p == 0,
LS2gen = concat( [Mod(bnf[8][4][2],bnf.pol)], LS2gen)); \\ LS2gen = concat( [Mod(bnf.tu[2],bnf.pol)], LS2gen));
diff --git a/src/sage/geometry/cone.py b/src/sage/geometry/cone.py
index 33b28dbede..1c6873eadd 100644
--- a/src/sage/geometry/cone.py
+++ b/src/sage/geometry/cone.py
@@ -4268,31 +4268,31 @@ class ConvexRationalPolyhedralCone(IntegralRayCollection, Container):
M(-5, 21, 0, -3),
M( 0, -2, 0, 1),
M(15, -63, 25, 9),
- M( 2, -3, 0, 1),
- M( 1, -4, 1, 1),
- M(-1, 3, 0, 0),
M( 4, -4, 0, 1),
- M( 1, -5, 2, 1),
M( 3, -5, 1, 1),
- M( 6, -5, 0, 1),
- M( 3, -13, 5, 2),
M( 2, -6, 2, 1),
- M( 5, -6, 1, 1),
- M( 0, 1, 0, 0),
- M( 8, -6, 0, 1),
+ M(-1, 7, 0, -1),
+ M( 6, -21, 8, 3),
+ M( 5, -21, 9, 3),
+ M(-1, 3, 0, 0),
+ M( 7, -28, 11, 4),
+ M( 1, -5, 2, 1),
M(-2, 8, 0, -1),
+ M( 8, -6, 0, 1),
+ M( 7, -7, 1, 1),
+ M( 3, -13, 5, 2),
+ M( 2, -3, 0, 1),
+ M( 1, -4, 1, 1),
+ M(-3, 14, 0, -2),
M(10, -42, 17, 6),
- M( 7, -28, 11, 4),
- M( 5, -21, 9, 3),
- M( 6, -21, 8, 3),
+ M( 1, 0, 0, 0),
+ M( 0, 0, 1, 0),
+ M( 6, -5, 0, 1),
+ M( 5, -6, 1, 1),
+ M( 4, -7, 2, 1),
M( 5, -14, 5, 2),
M( 2, -7, 3, 1),
- M( 4, -7, 2, 1),
- M( 7, -7, 1, 1),
- M( 0, 0, 1, 0),
- M(-3, 14, 0, -2),
- M(-1, 7, 0, -1),
- M( 1, 0, 0, 0)
+ M( 0, 1, 0, 0)
in 4-d lattice M
Not a strictly convex cone::
diff --git a/src/sage/groups/abelian_gps/abelian_group.py b/src/sage/groups/abelian_gps/abelian_group.py
index b61b94c7e4..f7ccab44a3 100644
--- a/src/sage/groups/abelian_gps/abelian_group.py
+++ b/src/sage/groups/abelian_gps/abelian_group.py
@@ -1451,7 +1451,7 @@ class AbelianGroup_class(UniqueRepresentation, AbelianGroupBase):
EXAMPLES::
sage: AbelianGroup([2,3]).subgroups()
- [Multiplicative Abelian subgroup isomorphic to C2 x C3 generated by {f0*f1^2},
+ [Multiplicative Abelian subgroup isomorphic to C2 x C3 generated by {f0*f1},
Multiplicative Abelian subgroup isomorphic to C2 generated by {f0},
Multiplicative Abelian subgroup isomorphic to C3 generated by {f1},
Trivial Abelian subgroup]
diff --git a/src/sage/groups/additive_abelian/additive_abelian_group.py b/src/sage/groups/additive_abelian/additive_abelian_group.py
index 3dcedeb7e3..48747e4870 100644
--- a/src/sage/groups/additive_abelian/additive_abelian_group.py
+++ b/src/sage/groups/additive_abelian/additive_abelian_group.py
@@ -61,18 +61,18 @@ def AdditiveAbelianGroup(invs, remember_generators = True):
((1, 0, 0), (0, 1, 0), (0, 0, 1))
sage: [H.0, H.1, H.2]
[(1, 0, 0), (0, 1, 0), (0, 0, 1)]
- sage: p=H.0+H.1+6*H.2; p
- (1, 1, 6)
+ sage: p=2*H.0+H.1+6*H.2; p
+ (2, 1, 6)
sage: H.smith_form_gens()
- ((2, 1, 0), (0, 0, 1))
+ ((1, 1, 0), (0, 0, 1))
sage: q=H.linear_combination_of_smith_form_gens([5,6]); q
- (1, 1, 6)
+ (2, 1, 6)
sage: p==q
True
- sage: r=H(vector([1,1,6])); r
- (1, 1, 6)
+ sage: r=H(vector([2,1,6])); r
+ (2, 1, 6)
sage: p==r
True
@@ -85,21 +85,21 @@ def AdditiveAbelianGroup(invs, remember_generators = True):
sage: G=AdditiveAbelianGroup([3,2,0], remember_generators=False)
sage: G.gens()
- ((2, 1, 0), (0, 0, 1))
+ ((1, 1, 0), (0, 0, 1))
sage: [G.0, G.1]
- [(2, 1, 0), (0, 0, 1)]
+ [(1, 1, 0), (0, 0, 1)]
sage: p=5*G.0+6*G.1; p
- (1, 1, 6)
+ (2, 1, 6)
sage: H.smith_form_gens()
- ((2, 1, 0), (0, 0, 1))
+ ((1, 1, 0), (0, 0, 1))
sage: q=G.linear_combination_of_smith_form_gens([5,6]); q
- (1, 1, 6)
+ (2, 1, 6)
sage: p==q
True
- sage: r=G(vector([1,1,6])); r
- (1, 1, 6)
+ sage: r=G(vector([2,1,6])); r
+ (2, 1, 6)
sage: p==r
True
@@ -427,7 +427,7 @@ class AdditiveAbelianGroup_fixed_gens(AdditiveAbelianGroup_class):
sage: G.gens()
((1, 0), (0, 1))
sage: G.smith_form_gens()
- ((1, 2),)
+ ((1, 1),)
"""
return self._orig_gens
diff --git a/src/sage/groups/fqf_orthogonal.py b/src/sage/groups/fqf_orthogonal.py
index 3c5190589b..c536f03064 100644
--- a/src/sage/groups/fqf_orthogonal.py
+++ b/src/sage/groups/fqf_orthogonal.py
@@ -156,9 +156,9 @@ class FqfOrthogonalGroup(AbelianGroupAutomorphismGroup_subgroup):
sage: S2 = 9 * T
sage: Q = S1/S2
sage: G = T.orthogonal_group()
- sage: g = G(matrix(ZZ, 2, [8, 0, 0, 1]))
+ sage: g = G(matrix(ZZ, 2, [7, 12, 8, 19]))
sage: Q.1 * g
- (0, 2)
+ (0, 1)
"""
Element = FqfIsometry
diff --git a/src/sage/lfunctions/pari.py b/src/sage/lfunctions/pari.py
index f810157b3e..f621b57c67 100644
--- a/src/sage/lfunctions/pari.py
+++ b/src/sage/lfunctions/pari.py
@@ -421,7 +421,7 @@ class LFunction(SageObject):
sage: L.derivative(1,E.rank())
1.51863300057685
sage: L.taylor_series(1,4)
- -3...e-19 + (...e-19)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4)
+ ...e-19 + (...e-19)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4)
.. RUBRIC:: Number field
diff --git a/src/sage/libs/pari/__init__.py b/src/sage/libs/pari/__init__.py
index 77eda66097..3fa4618631 100644
--- a/src/sage/libs/pari/__init__.py
+++ b/src/sage/libs/pari/__init__.py
@@ -161,12 +161,12 @@ exact object. Therefore, you should set the precision for each method
call individually::
sage: e = pari([0,0,0,-82,0]).ellinit()
- sage: eta1 = e.elleta(precision=100)[0]
+ sage: eta1 = e.elleta(precision=50)[0]
sage: eta1.sage()
3.6054636014326520859158205642077267748
sage: eta1 = e.elleta(precision=180)[0]
sage: eta1.sage()
- 3.60546360143265208591582056420772677481026899659802474544
+ 3.605463601432652085915820564207726774810268996598024745444380641429820491740
"""
diff --git a/src/sage/libs/pari/convert_sage.pyx b/src/sage/libs/pari/convert_sage.pyx
index 33aa4d2d46..b506227cff 100644
--- a/src/sage/libs/pari/convert_sage.pyx
+++ b/src/sage/libs/pari/convert_sage.pyx
@@ -144,6 +144,20 @@ cpdef gen_to_sage(Gen z, locals=None):
sage: a.parent()
Complex Field with 64 bits of precision
+ sage: z = pari('1 + 1.0*I'); z
+ 1 + 1.00000000000000*I
+ sage: a = gen_to_sage(z); a
+ 1.00000000000000000 + 1.00000000000000000*I
+ sage: a.parent()
+ Complex Field with 64 bits of precision
+
+ sage: z = pari('1.0 + 1*I'); z
+ 1.00000000000000 + I
+ sage: a = gen_to_sage(z); a
+ 1.00000000000000000 + 1.00000000000000000*I
+ sage: a.parent()
+ Complex Field with 64 bits of precision
+
Converting polynomials::
sage: f = pari('(2/3)*x^3 + x - 5/7 + y')
@@ -241,7 +255,9 @@ cpdef gen_to_sage(Gen z, locals=None):
elif t == t_FRAC:
return Rational(z)
elif t == t_REAL:
- prec = prec_words_to_bits(z.precision())
+ prec = z.bitprecision()
+ if prec.type() == 't_INFINITY':
+ prec = 53
return RealField(prec)(z)
elif t == t_COMPLEX:
real = z.real()
@@ -251,14 +267,19 @@ cpdef gen_to_sage(Gen z, locals=None):
if tx in [t_INTMOD, t_PADIC] or ty in [t_INTMOD, t_PADIC]:
raise NotImplementedError("No conversion to python available for t_COMPLEX with t_INTMOD or t_PADIC components")
if tx == t_REAL or ty == t_REAL:
- xprec = real.precision() # will be 0 if exact
- yprec = imag.precision() # will be 0 if exact
- if xprec == 0:
- prec = prec_words_to_bits(yprec)
- elif yprec == 0:
- prec = prec_words_to_bits(xprec)
+ xprec = real.bitprecision() # will be 0 if exact
+ yprec = imag.bitprecision() # will be 0 if exact
+ if xprec == 0 or yprec == 0:
+ raise RuntimeError
+ if xprec.type() == 't_INFINITY':
+ if yprec.type() == 't_INFINITY':
+ prec = 53
+ else:
+ prec = yprec
+ elif yprec.type() == 't_INFINITY':
+ prec = xprec
else:
- prec = max(prec_words_to_bits(xprec), prec_words_to_bits(yprec))
+ prec = max(xprec, yprec)
R = RealField(prec)
C = ComplexField(prec)
diff --git a/src/sage/libs/pari/tests.py b/src/sage/libs/pari/tests.py
index eb1c04f1e6..40329aea82 100644
--- a/src/sage/libs/pari/tests.py
+++ b/src/sage/libs/pari/tests.py
@@ -135,7 +135,7 @@ Some more exotic examples::
sage: K. = NumberField(polygen(QQ)^3 - 2)
sage: pari(K)
- [y^3 - 2, [1, 1], -108, 1, [[1, 1.25992104989487, 1.58740105196820; 1, -0.629960524947437 + 1.09112363597172*I, -0.793700525984100 - 1.37472963699860*I], [1, 1.25992104989487, 1.58740105196820; 1, 0.461163111024285, -2.16843016298270; 1, -1.72108416091916, 0.581029111014503], [1, 1, 2; 1, 0, -2; 1, -2, 1], [3, 0, 0; 0, 0, 6; 0, 6, 0], [6, 0, 0; 0, 6, 0; 0, 0, 3], [2, 0, 0; 0, 0, 1; 0, 1, 0], [2, [0, 0, 2; 1, 0, 0; 0, 1, 0]], []], [1.25992104989487, -0.629960524947437 + 1.09112363597172*I], [1, y, y^2], [1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 0, 0, 0, 0, 2, 0, 2, 0; 0, 1, 0, 1, 0, 0, 0, 0, 2; 0, 0, 1, 0, 1, 0, 1, 0, 0]]
+ [y^3 - 2, [1, 1], -108, 1, [[1, 1.25992104989487, 1.58740105196820; 1, -0.629960524947437 + 1.09112363597172*I, -0.793700525984100 - 1.37472963699860*I], [1, 1.25992104989487, 1.58740105196820; 1, 0.461163111024285, -2.16843016298270; 1, -1.72108416091916, 0.581029111014503], [16, 20, 25; 16, 7, -35; 16, -28, 9], [3, 0, 0; 0, 0, 6; 0, 6, 0], [6, 0, 0; 0, 6, 0; 0, 0, 3], [2, 0, 0; 0, 0, 1; 0, 1, 0], [2, [0, 0, 2; 1, 0, 0; 0, 1, 0]], [2, 3]], [1.25992104989487, -0.629960524947437 + 1.09112363597172*I], [1, y, y^2], [1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 0, 0, 0, 0, 2, 0, 2, 0; 0, 1, 0, 1, 0, 0, 0, 0, 2; 0, 0, 1, 0, 1, 0, 1, 0, 0]]
sage: E = EllipticCurve('37a1')
sage: pari(E)
@@ -375,13 +375,13 @@ Constructors::
sage: pari('["bc","ab","bc"]').Set()
["ab", "bc"]
- sage: pari([65,66,123]).Strchr()
+ sage: pari([65,66,123]).strchr()
"AB{"
sage: pari('"Sage"').Vecsmall()
Vecsmall([83, 97, 103, 101])
- sage: _.Strchr()
+ sage: _.strchr()
"Sage"
- sage: pari([83, 97, 103, 101]).Strchr()
+ sage: pari([83, 97, 103, 101]).strchr()
"Sage"
Basic functions::
@@ -448,7 +448,7 @@ Basic functions::
sage: pari('x').component(0)
Traceback (most recent call last):
...
- PariError: non-existent component: index < 1
+ PariError: nonexistent component: index < 1
sage: pari('x+1').conj()
x + 1
@@ -767,7 +767,7 @@ Transcendental functions::
sage: pari(2).besseli(3+i)
1.12539407613913 + 2.08313822670661*I
sage: C. = ComplexField()
- sage: pari(2+i).besseln(3)
+ sage: pari(2+i).bessely(3)
-0.280775566958244 - 0.486708533223726*I
sage: pari(1.5).cos()
@@ -822,7 +822,7 @@ Transcendental functions::
sage: pari(-1).gamma()
Traceback (most recent call last):
...
- PariError: domain error in gamma: argument = non-positive integer
+ PariError: domain error in gamma: argument = nonpositive integer
sage: pari(2).gammah()
1.32934038817914
@@ -1633,7 +1633,7 @@ General number fields::
sage: x = QQ['x'].0; nf = pari(x^2 + 2).nfinit()
sage: nf.nfgaloisconj()
- [x, -x]~
+ [-x, x]~
sage: nf = pari(x^3 + 2).nfinit()
sage: nf.nfgaloisconj()
[x]~
@@ -1676,7 +1676,7 @@ General number fields::
[[1, [7605, 4]~, [5610, 5]~, [7913, -6]~; 0, 1, 0, -1; 0, 0, 1, 0; 0, 0, 0, 1], [[19320, 13720; 0, 56], [2, 1; 0, 1], 1, 1]]
sage: pari('x^3 - 17').nfinit()
- [x^3 - 17, [1, 1], -867, 3, [[1, 1.68006914259990, 2.57128159065824; 1, -0.340034571299952 - 2.65083754153991*I, -1.28564079532912 + 2.22679517779329*I], [1, 1.68006914259990, 2.57128159065824; 1, -2.99087211283986, 0.941154382464174; 1, 2.31080297023995, -3.51243597312241], [1, 2, 3; 1, -3, 1; 1, 2, -4], [3, 1, 0; 1, -11, 17; 0, 17, 0], [51, 0, 16; 0, 17, 3; 0, 0, 1], [17, 0, -1; 0, 0, 3; -1, 3, 2], [51, [-17, 6, -1; 0, -18, 3; 1, 0, -16]], [3, 17]], [2.57128159065824, -1.28564079532912 + 2.22679517779329*I], [3, x^2 - x + 1, 3*x], [1, 0, -1; 0, 0, 3; 0, 1, 1], [1, 0, 0, 0, -4, 6, 0, 6, -1; 0, 1, 0, 1, 1, -1, 0, -1, 3; 0, 0, 1, 0, 2, 0, 1, 0, 1]]
+ [x^3 - 17, [1, 1], -867, 3, [[1, 1.68006914259990, 2.57128159065824; 1, -0.340034571299952 - 2.65083754153991*I, -1.28564079532912 + 2.22679517779329*I], [1, 1.68006914259990, 2.57128159065824; 1, -2.99087211283986, 0.941154382464174; 1, 2.31080297023995, -3.51243597312241], [16, 27, 41; 16, -48, 15; 16, 37, -56], [3, 1, 0; 1, -11, 17; 0, 17, 0], [51, 0, 16; 0, 17, 3; 0, 0, 1], [17, 0, -1; 0, 0, 3; -1, 3, 2], [51, [-17, 6, -1; 0, -18, 3; 1, 0, -16]], [3, 17]], [2.57128159065824, -1.28564079532912 + 2.22679517779329*I], [3, x^2 - x + 1, 3*x], [1, 0, -1; 0, 0, 3; 0, 1, 1], [1, 0, 0, 0, -4, 6, 0, 6, -1; 0, 1, 0, 1, 1, -1, 0, -1, 3; 0, 0, 1, 0, 2, 0, 1, 0, 1]]
sage: pari('x^2 + 10^100 + 1').nfinit()
[...]
sage: pari('1.0').nfinit()
@@ -1737,7 +1737,7 @@ General number fields::
sage: pari(-23).quadhilbert()
x^3 - x^2 + 1
sage: pari(145).quadhilbert()
- x^4 - 6*x^2 - 5*x - 1
+ x^4 - x^3 - 5*x^2 - x + 1
sage: pari(-12).quadhilbert() # Not fundamental
Traceback (most recent call last):
...
@@ -1762,7 +1762,7 @@ library::
sage: e = pari([0,0,0,-82,0]).ellinit()
sage: eta1 = e.elleta(precision=100)[0]
sage: eta1.sage()
- 3.6054636014326520859158205642077267748
+ 3.60546360143265208591582056420772677481026899659802474544
sage: eta1 = e.elleta(precision=180)[0]
sage: eta1.sage()
3.60546360143265208591582056420772677481026899659802474544
diff --git a/src/sage/matrix/matrix1.pyx b/src/sage/matrix/matrix1.pyx
index 0455c169fe..4fa3cb9b2d 100644
--- a/src/sage/matrix/matrix1.pyx
+++ b/src/sage/matrix/matrix1.pyx
@@ -87,7 +87,7 @@ cdef class Matrix(Matrix0):
[1.000000000, 2.000000000; 3.000000000, 1.000000000] # 32-bit
[1.00000000000000, 2.00000000000000; 3.00000000000000, 1.00000000000000] # 64-bit
sage: b[0][0].precision() # in words
- 3
+ 19
"""
from sage.libs.pari.all import pari
return pari.matrix(self._nrows, self._ncols, self._list())
diff --git a/src/sage/modular/local_comp/liftings.py b/src/sage/modular/local_comp/liftings.py
index 91bfd2448d..f9bf800b22 100644
--- a/src/sage/modular/local_comp/liftings.py
+++ b/src/sage/modular/local_comp/liftings.py
@@ -222,9 +222,9 @@ def lift_for_SL(A, N=None):
TESTS::
sage: lift_for_SL(matrix(3,3,[1,2,0,3,4,0,0,0,1]),3)
- [10 14 3]
- [ 9 10 3]
- [ 3 3 1]
+ [ -2 -4 3]
+ [ -9 -14 3]
+ [ -6 -9 1]
sage: A = matrix(Zmod(7), 2, [1,0,0,1])
sage: L = lift_for_SL(A)
diff --git a/src/sage/modular/local_comp/smoothchar.py b/src/sage/modular/local_comp/smoothchar.py
index 6dedb26e7e..3b1ac90460 100644
--- a/src/sage/modular/local_comp/smoothchar.py
+++ b/src/sage/modular/local_comp/smoothchar.py
@@ -1617,8 +1617,8 @@ class SmoothCharacterGroupRamifiedQuadratic(SmoothCharacterGroupGeneric):
sage: G = SmoothCharacterGroupRamifiedQuadratic(3, 1, QQ)
sage: s = G.number_field().gen()
sage: G.discrete_log(4, 3 + 2*s)
- [5, 1, 1, 1]
- sage: gs = G.unit_gens(4); gs[0]^5 * gs[1] * gs[2] * gs[3] - (3 + 2*s) in G.ideal(4)
+ [1, 2, 2, 1]
+ sage: gs = G.unit_gens(4); gs[0] * gs[1]^2 * gs[2]^2 * gs[3] - (3 + 2*s) in G.ideal(4)
True
"""
x = self.number_field().coerce(x)
diff --git a/src/sage/modular/modsym/p1list_nf.py b/src/sage/modular/modsym/p1list_nf.py
index 7a91d353e3..b71fc8aac4 100644
--- a/src/sage/modular/modsym/p1list_nf.py
+++ b/src/sage/modular/modsym/p1list_nf.py
@@ -956,7 +956,7 @@ class P1NFList(SageObject):
sage: N = k.ideal(a + 1)
sage: P = P1NFList(N)
sage: u = k.unit_group().gens_values(); u
- [-1, a^3 + a^2 + a + 12, a^3 + 3*a^2 - 1]
+ [-1, -a^3 - a^2 - a - 12, -a^3 - 3*a^2 + 1]
sage: P.apply_J_epsilon(3, u[2]^2)==P.apply_J_epsilon(P.apply_J_epsilon(3, u[2]),u[2])
True
"""
diff --git a/src/sage/modular/multiple_zeta.py b/src/sage/modular/multiple_zeta.py
index d42be82bd9..5f0057474c 100644
--- a/src/sage/modular/multiple_zeta.py
+++ b/src/sage/modular/multiple_zeta.py
@@ -457,7 +457,7 @@ class MultizetaValues(UniqueRepresentation):
"""
self.prec = int(prec)
self.max_weight = int(max_weight)
- self._data = pari.zetamultall(self.max_weight, self.prec)
+ self._data = pari.zetamultall(self.max_weight, precision=self.prec)
def update(self, max_weight, prec):
"""
diff --git a/src/sage/modules/fg_pid/fgp_element.py b/src/sage/modules/fg_pid/fgp_element.py
index 53857d31a0..6557b7f308 100644
--- a/src/sage/modules/fg_pid/fgp_element.py
+++ b/src/sage/modules/fg_pid/fgp_element.py
@@ -39,7 +39,7 @@ class FGP_Element(ModuleElement):
sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2])
sage: Q = V/W
sage: x = Q(V.0-V.1); x #indirect doctest
- (0, 3)
+ (0, 9)
sage: isinstance(x, sage.modules.fg_pid.fgp_element.FGP_Element)
True
sage: type(x)
@@ -94,14 +94,14 @@ class FGP_Element(ModuleElement):
sage: Q.1
(0, 1)
sage: Q.0.lift()
- (0, 0, 1)
+ (0, -6, 1)
sage: Q.1.lift()
- (0, 2, 0)
+ (0, -2, 0)
sage: x = Q(V.0); x
- (0, 4)
+ (0, 8)
sage: x.lift()
(1/2, 0, 0)
- sage: x == 4*Q.1
+ sage: x == 8*Q.1
True
sage: x.lift().parent() == V
True
@@ -158,9 +158,9 @@ class FGP_Element(ModuleElement):
We test canonical coercion from V and W.
sage: Q.0 + V.0
- (1, 4)
+ (1, 8)
sage: V.0 + Q.0
- (1, 4)
+ (1, 8)
sage: W.0 + Q.0
(1, 0)
sage: W.0 + Q.0 == Q.0
@@ -291,7 +291,7 @@ class FGP_Element(ModuleElement):
sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2])
sage: Q = V/W
sage: Q(V.1)._repr_()
- '(0, 1)'
+ '(0, 11)'
"""
return repr(self.vector())
diff --git a/src/sage/modules/fg_pid/fgp_module.py b/src/sage/modules/fg_pid/fgp_module.py
index 35f7cdaa17..5a700d808e 100644
--- a/src/sage/modules/fg_pid/fgp_module.py
+++ b/src/sage/modules/fg_pid/fgp_module.py
@@ -70,17 +70,17 @@ the technical note has a V that need not be equal to V0, in general. ::
sage: M0.optimized()[0].V()
Free module of degree 3 and rank 2 over Integer Ring
User basis matrix:
- [0 0 1]
- [0 2 0]
+ [ 0 -8 1]
+ [ 0 -2 0]
Create elements of M0 either by coercing in elements of V0, getting generators,
or coercing in a list or tuple or coercing in 0. Finally, one can express an
element as a linear combination of the smith form generators ::
sage: M0(V0.0)
- (0, 14)
+ (0, 2)
sage: M0(V0.0 + W0.0) # no difference modulo W0
- (0, 14)
+ (0, 2)
sage: M0.linear_combination_of_smith_form_gens([3,20])
(3, 4)
sage: 3*M0.0 + 20*M0.1
@@ -93,9 +93,9 @@ coerces to V0, then take the equivalence class modulo W0. ::
sage: x = M0.0 - M0.1; x
(1, 15)
sage: x.lift()
- (0, -2, 1)
+ (0, -6, 1)
sage: M0(vector([1/2,0,0]))
- (0, 14)
+ (0, 2)
sage: x.additive_order()
16
@@ -144,7 +144,7 @@ You can explicitly coerce elements of the kernel into M0 though. ::
sage: M0(K.0)
(2, 0)
sage: M0(K.1)
- (3, 1)
+ (1, 13)
sage: f(M0(K.0))
(0)
sage: f(M0(K.1))
@@ -180,7 +180,7 @@ TESTS::
sage: Q.linear_combination_of_smith_form_gens([1,3])
(1, 3)
sage: Q(V([1,3,4]))
- (0, 11)
+ (0, 1)
sage: Q(W([1,16,0]))
(0, 0)
sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],QQ)
@@ -632,7 +632,7 @@ class FGP_Module_class(Module):
sage: W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2])
sage: Q = V/W
sage: x = Q(V.0-V.1); x # indirect doctest
- (0, 3)
+ (0, 9)
sage: type(x)
sage: x is Q(x)
@@ -931,9 +931,9 @@ class FGP_Module_class(Module):
sage: Q = V/W
sage: Q._smith_form()
(
- [ 1 0 0] [1 0 0] [ 1 0 -8]
- [ 0 4 0] [0 0 1] [ 0 0 1]
- [ 0 0 12], [0 1 0], [ 0 1 0]
+ [ 1 0 0] [ 1 0 1] [ 1 -4 20]
+ [ 0 4 0] [ 0 -1 1] [ 0 0 -1]
+ [ 0 0 12], [ 0 -1 0], [ 0 1 -3]
)
"""
return self._relative_matrix().smith_form()
@@ -1026,7 +1026,7 @@ class FGP_Module_class(Module):
sage: Q.smith_form_gens()
((1, 0), (0, 1))
sage: [x.lift() for x in Q.smith_form_gens()]
- [(0, 0, 1), (0, 1, 0)]
+ [(0, -3, 1), (0, -1, 0)]
"""
# Get the rightmost transformation in the Smith form
_, _, X = self._smith_form()
@@ -1067,18 +1067,18 @@ class FGP_Module_class(Module):
[ 0 0 0 1/3 0]
[ 0 0 0 0 2/3]
sage: D.gens_to_smith()
- [0 3 0]
+ [0 3 6]
[0 0 3]
- [0 2 0]
- [1 0 0]
- [0 0 4]
+ [0 4 4]
+ [1 2 0]
+ [0 0 8]
sage: T = D.gens_to_smith()*D.smith_to_gens()
sage: T
- [ 3 0 15 0 0]
- [ 0 33 0 0 3]
- [ 2 0 10 0 0]
- [ 0 0 0 1 0]
- [ 0 44 0 0 4]
+ [27 48 3 0 60]
+ [12 21 0 0 24]
+ [20 36 4 0 48]
+ [ 2 4 3 1 9]
+ [32 56 0 0 64]
The matrix `T` now satisfies a certain congruence::
@@ -1120,14 +1120,14 @@ class FGP_Module_class(Module):
[ 0 0 0 1/3 0]
[ 0 0 0 0 2/3]
sage: D.smith_to_gens()
- [ 0 0 0 1 0]
- [ 1 0 5 0 0]
- [ 0 11 0 0 1]
+ [0 0 1 1 1]
+ [1 2 1 0 4]
+ [4 7 0 0 8]
sage: T = D.smith_to_gens()*D.gens_to_smith()
sage: T
- [ 1 0 0]
- [ 0 13 0]
- [ 0 0 37]
+ [ 1 6 12]
+ [ 0 7 48]
+ [ 0 12 109]
This matrix satisfies the congruence::
@@ -1148,7 +1148,7 @@ class FGP_Module_class(Module):
of the user defined generators that is x::
sage: x.vector() * D.smith_to_gens()
- (2, 33, 10, 1, 3)
+ (14, 25, 3, 1, 33)
"""
if self.base_ring() != ZZ:
# it is not
@@ -1196,7 +1196,7 @@ class FGP_Module_class(Module):
sage: gens = [V(g) for g in gens]
sage: D = FGP_with_gens(V, W, gens)
sage: D.gens()
- ((0, 3, 0), (0, 0, 3), (0, 2, 0), (1, 0, 0), (0, 0, 8))
+ ((0, 3, 6), (0, 0, 9), (0, 4, 4), (1, 2, 0), (0, 0, 4))
We create some element of D::
@@ -1209,12 +1209,12 @@ class FGP_Module_class(Module):
sage: v = D.gens_vector(x)
sage: v
- (2, 9, 10, 1, 33)
+ (26, 11, 3, 1, 18)
The output can be further reduced::
sage: D.gens_vector(x, reduce=True)
- (0, 1, 1, 1, 0)
+ (0, 3, 0, 1, 0)
Let us check::
@@ -1278,28 +1278,28 @@ class FGP_Module_class(Module):
sage: O.V()
Free module of degree 3 and rank 2 over Integer Ring
User basis matrix:
- [0 0 1]
- [0 2 0]
+ [ 0 -6 1]
+ [ 0 -2 0]
sage: phi = Q.hom([Q.0, 4*Q.1])
sage: x = Q(V.0); x
- (0, 4)
+ (0, 8)
sage: Q.coordinate_vector(x, reduce=True)
- (0, 4)
+ (0, 8)
sage: Q.coordinate_vector(-x, reduce=False) # random
- (0, -4)
- sage: x == 4*Q.1
+ (0, -8)
+ sage: x == 8*Q.1
True
sage: x = Q(V.1); x
- (0, 1)
+ (0, 11)
sage: Q.coordinate_vector(x)
- (0, 1)
- sage: x == Q.1
+ (0, -1)
+ sage: x == -Q.1
True
sage: x = Q(V.2); x
- (1, 0)
+ (1, 9)
sage: Q.coordinate_vector(x)
- (1, 0)
- sage: x == Q.0
+ (1, -3)
+ sage: x == Q.0-3*Q.1
True
"""
try:
@@ -1407,8 +1407,8 @@ class FGP_Module_class(Module):
sage: O.V()
Free module of degree 3 and rank 2 over Integer Ring
User basis matrix:
- [0 0 1]
- [0 1 0]
+ [ 0 -3 1]
+ [ 0 -1 0]
sage: O.W()
Free module of degree 3 and rank 2 over Integer Ring
Echelon basis matrix:
@@ -1699,7 +1699,7 @@ class FGP_Module_class(Module):
sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2])
sage: Q = V/W
sage: Q.random_element()
- (1, 10)
+ (1, 11)
"""
return self(self._V.random_element(*args, **kwds))
diff --git a/src/sage/modules/fg_pid/fgp_morphism.py b/src/sage/modules/fg_pid/fgp_morphism.py
index fbbd479c6e..679965df96 100644
--- a/src/sage/modules/fg_pid/fgp_morphism.py
+++ b/src/sage/modules/fg_pid/fgp_morphism.py
@@ -258,20 +258,20 @@ class FGP_Morphism(Morphism):
sage: O.V()
Free module of degree 3 and rank 2 over Integer Ring
User basis matrix:
- [0 0 1]
- [0 2 0]
+ [ 0 -6 1]
+ [ 0 -2 0]
sage: phi = Q.hom([Q.0, 4*Q.1])
sage: x = Q(V.0); x
- (0, 4)
- sage: x == 4*Q.1
+ (0, 8)
+ sage: x == 8*Q.1
True
sage: x in O.V()
False
sage: phi(x)
- (0, 4)
+ (0, 8)
sage: phi(4*Q.1)
(0, 4)
- sage: phi(4*Q.1) == phi(x)
+ sage: phi(8*Q.1) == phi(x)
True
"""
from .fgp_module import is_FGP_Module
diff --git a/src/sage/modules/torsion_quadratic_module.py b/src/sage/modules/torsion_quadratic_module.py
index 39e7065ac4..6086d1180d 100644
--- a/src/sage/modules/torsion_quadratic_module.py
+++ b/src/sage/modules/torsion_quadratic_module.py
@@ -1230,24 +1230,24 @@ class TorsionQuadraticModule(FGP_Module_class, CachedRepresentation):
sage: q.twist(-1)
Finite quadratic module over Integer Ring with invariants (3, 9)
Gram matrix of the quadratic form with values in Q/Z:
- [2/3 0]
- [ 0 8/9]
+ [2/3 1/3]
+ [1/3 8/9]
This form is defined modulo `3`::
sage: q.twist(3)
Finite quadratic module over Integer Ring with invariants (3, 9)
Gram matrix of the quadratic form with values in Q/3Z:
- [ 1 0]
- [ 0 1/3]
+ [ 1 2]
+ [ 2 1/3]
The next form is defined modulo `4`::
sage: q.twist(4)
Finite quadratic module over Integer Ring with invariants (3, 9)
Gram matrix of the quadratic form with values in Q/4Z:
- [4/3 0]
- [ 0 4/9]
+ [4/3 8/3]
+ [8/3 4/9]
"""
s = self.base_ring().fraction_field()(s)
n = self.V().degree()
diff --git a/src/sage/quadratic_forms/genera/genus.py b/src/sage/quadratic_forms/genera/genus.py
index 2e773616e2..0daa55f84f 100644
--- a/src/sage/quadratic_forms/genera/genus.py
+++ b/src/sage/quadratic_forms/genera/genus.py
@@ -2897,18 +2897,18 @@ class GenusSymbol_global_ring(object):
sage: GS.discriminant_form()
Finite quadratic module over Integer Ring with invariants (2, 2, 4, 24)
Gram matrix of the quadratic form with values in Q/2Z:
- [ 1/2 0 0 0]
- [ 0 3/2 0 0]
- [ 0 0 7/4 0]
- [ 0 0 0 7/24]
+ [ 1/2 0 0 0]
+ [ 0 1/2 1/2 0]
+ [ 0 1/2 7/4 0]
+ [ 0 0 0 31/24]
sage: A = matrix.diagonal(ZZ, [1, -4, 6, 8])
sage: GS = Genus(A)
sage: GS.discriminant_form()
Finite quadratic module over Integer Ring with invariants (2, 4, 24)
Gram matrix of the quadratic form with values in Q/Z:
- [ 1/2 0 0]
- [ 0 3/4 0]
- [ 0 0 7/24]
+ [ 1/2 1/2 0]
+ [ 1/2 3/4 0]
+ [ 0 0 19/24]
"""
from sage.modules.torsion_quadratic_module import TorsionQuadraticForm
qL = []
diff --git a/src/sage/rings/finite_rings/finite_field_constructor.py b/src/sage/rings/finite_rings/finite_field_constructor.py
index 81ff16b1b6..46d56b923f 100644
--- a/src/sage/rings/finite_rings/finite_field_constructor.py
+++ b/src/sage/rings/finite_rings/finite_field_constructor.py
@@ -284,11 +284,6 @@ class FiniteFieldFactory(UniqueFactory):
(a generator of the multiplicative group), use
``modulus="primitive"`` if you need this::
- sage: K. = GF(5^40)
- sage: a.multiplicative_order()
- 189478062869360049565633138
- sage: a.is_square()
- True
sage: K. = GF(5^40, modulus="primitive")
sage: b.multiplicative_order()
9094947017729282379150390624
diff --git a/src/sage/rings/finite_rings/integer_mod_ring.py b/src/sage/rings/finite_rings/integer_mod_ring.py
index 0dcef0d21a..7b4f8b1b4b 100644
--- a/src/sage/rings/finite_rings/integer_mod_ring.py
+++ b/src/sage/rings/finite_rings/integer_mod_ring.py
@@ -626,7 +626,7 @@ class IntegerModRing_generic(quotient_ring.QuotientRing_generic):
sage: Integers(5).multiplicative_subgroups()
((2,), (4,), ())
sage: Integers(15).multiplicative_subgroups()
- ((11, 7), (4, 11), (8,), (11,), (14,), (7,), (4,), ())
+ ((14, 13), (4, 11), (8,), (11,), (14,), (7,), (4,), ())
sage: Integers(2).multiplicative_subgroups()
((),)
sage: len(Integers(341).multiplicative_subgroups())
diff --git a/src/sage/rings/finite_rings/residue_field.pyx b/src/sage/rings/finite_rings/residue_field.pyx
index cd4c2212c3..007a68e4c0 100644
--- a/src/sage/rings/finite_rings/residue_field.pyx
+++ b/src/sage/rings/finite_rings/residue_field.pyx
@@ -1055,7 +1055,7 @@ cdef class ReductionMap(Map):
sage: f = k.convert_map_from(K)
sage: s = f.section(); s
Lifting map:
- From: Residue field in abar of Fractional ideal (14*a^4 - 24*a^3 - 26*a^2 + 58*a - 15)
+ From: Residue field in abar of Fractional ideal (-14*a^4 + 24*a^3 + 26*a^2 - 58*a + 15)
To: Number Field in a with defining polynomial x^5 - 5*x + 2
sage: s(k.gen())
a
@@ -1268,7 +1268,7 @@ cdef class ResidueFieldHomomorphism_global(RingHomomorphism):
sage: f = k.coerce_map_from(K.ring_of_integers())
sage: s = f.section(); s
Lifting map:
- From: Residue field in abar of Fractional ideal (14*a^4 - 24*a^3 - 26*a^2 + 58*a - 15)
+ From: Residue field in abar of Fractional ideal (-14*a^4 + 24*a^3 + 26*a^2 - 58*a + 15)
To: Maximal Order in Number Field in a with defining polynomial x^5 - 5*x + 2
sage: s(k.gen())
a
@@ -1371,10 +1371,10 @@ cdef class LiftingMap(Section):
sage: F = K.factor(7)[0][0].residue_field()
sage: L = F.lift_map(); L
Lifting map:
- From: Residue field in abar of Fractional ideal (-2*a^4 + a^3 - 4*a^2 + 2*a - 1)
+ From: Residue field in abar of Fractional ideal (2*a^4 - a^3 + 4*a^2 - 2*a + 1)
To: Maximal Order in Number Field in a with defining polynomial x^5 + 2
sage: L.domain()
- Residue field in abar of Fractional ideal (-2*a^4 + a^3 - 4*a^2 + 2*a - 1)
+ Residue field in abar of Fractional ideal (2*a^4 - a^3 + 4*a^2 - 2*a + 1)
sage: K. = CyclotomicField(7)
sage: F = K.factor(5)[0][0].residue_field()
@@ -1498,7 +1498,7 @@ cdef class LiftingMap(Section):
sage: F. = K.factor(7)[0][0].residue_field()
sage: F.lift_map() #indirect doctest
Lifting map:
- From: Residue field in tmod of Fractional ideal (-3*theta_12^2 + 1)
+ From: Residue field in tmod of Fractional ideal (theta_12^2 + 2)
To: Maximal Order in Cyclotomic Field of order 12 and degree 4
"""
return "Lifting"
@@ -1515,7 +1515,7 @@ class ResidueFiniteField_prime_modn(ResidueField_generic, FiniteField_prime_modn
sage: P = K.ideal(29).factor()[1][0]
sage: k = ResidueField(P)
sage: k
- Residue field of Fractional ideal (a^2 + 2*a + 2)
+ Residue field of Fractional ideal (-a^2 - 2*a - 2)
sage: k.order()
29
sage: OK = K.maximal_order()
@@ -1597,7 +1597,7 @@ class ResidueFiniteField_prime_modn(ResidueField_generic, FiniteField_prime_modn
sage: P = K.ideal(29).factor()[1][0]
sage: k = ResidueField(P)
sage: k
- Residue field of Fractional ideal (a^2 + 2*a + 2)
+ Residue field of Fractional ideal (-a^2 - 2*a - 2)
sage: OK = K.maximal_order()
sage: c = OK(a)
sage: b = k(a); b
diff --git a/src/sage/rings/integer.pyx b/src/sage/rings/integer.pyx
index 77a3a18913..421a90e10e 100644
--- a/src/sage/rings/integer.pyx
+++ b/src/sage/rings/integer.pyx
@@ -5474,7 +5474,7 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
sage: 3._bnfisnorm(QuadraticField(-1, 'i'))
(1, 3)
sage: 7._bnfisnorm(CyclotomicField(7))
- (-zeta7^5 - zeta7^4 - 2*zeta7^3 - zeta7^2 - zeta7 - 1, 1)
+ (zeta7^5 - zeta7^2, 1)
"""
from sage.rings.rational_field import QQ
return QQ(self)._bnfisnorm(K, proof=proof, extra_primes=extra_primes)
diff --git a/src/sage/rings/number_field/S_unit_solver.py b/src/sage/rings/number_field/S_unit_solver.py
index 3d6601d536..ba8b5164f0 100644
--- a/src/sage/rings/number_field/S_unit_solver.py
+++ b/src/sage/rings/number_field/S_unit_solver.py
@@ -24,10 +24,10 @@ EXAMPLES::
sage: from sage.rings.number_field.S_unit_solver import solve_S_unit_equation, eq_up_to_order
sage: K. = NumberField(x^2+x+1)
sage: S = K.primes_above(3)
- sage: expected = [((2, 1), (4, 0), xi + 2, -xi - 1),
- ....: ((5, -1), (4, -1), 1/3*xi + 2/3, -1/3*xi + 1/3),
- ....: ((5, 0), (1, 0), -xi, xi + 1),
- ....: ((1, 1), (2, 0), -xi + 1, xi)]
+ sage: expected = [((0, 1), (4, 0), xi + 2, -xi - 1),
+ ....: ((1, -1), (0, -1), 1/3*xi + 2/3, -1/3*xi + 1/3),
+ ....: ((1, 0), (5, 0), xi + 1, -xi),
+ ....: ((2, 0), (5, 1), xi, -xi + 1)]
sage: sols = solve_S_unit_equation(K, S, 200)
sage: eq_up_to_order(sols, expected)
True
@@ -1780,20 +1780,20 @@ def sieve_ordering(SUK, q):
sage: SUK = K.S_unit_group(S=3)
sage: sieve_data = list(sieve_ordering(SUK, 19))
sage: sieve_data[0]
- (Fractional ideal (-2*xi^2 + 3),
- Fractional ideal (xi - 3),
- Fractional ideal (2*xi + 1))
+ (Fractional ideal (xi - 3),
+ Fractional ideal (-2*xi^2 + 3),
+ Fractional ideal (2*xi + 1))
sage: sieve_data[1]
- (Residue field of Fractional ideal (-2*xi^2 + 3),
- Residue field of Fractional ideal (xi - 3),
- Residue field of Fractional ideal (2*xi + 1))
+ (Residue field of Fractional ideal (xi - 3),
+ Residue field of Fractional ideal (-2*xi^2 + 3),
+ Residue field of Fractional ideal (2*xi + 1))
sage: sieve_data[2]
- ([18, 9, 16, 8], [18, 7, 10, 4], [18, 3, 12, 10])
+ ([18, 7, 16, 4], [18, 9, 12, 8], [18, 3, 10, 10])
sage: sieve_data[3]
- (972, 972, 3888)
+ (486, 648, 11664)
"""
K = SUK.number_field()
@@ -2654,10 +2654,10 @@ def sieve_below_bound(K, S, bound=10, bump=10, split_primes_list=[], verbose=Fal
sage: S = SUK.primes()
sage: sols = sieve_below_bound(K, S, 10)
sage: expected = [
- ....: ((5, -1), (4, -1), 1/3*xi + 2/3, -1/3*xi + 1/3),
- ....: ((2, 1), (4, 0), xi + 2, -xi - 1),
- ....: ((2, 0), (1, 1), xi, -xi + 1),
- ....: ((5, 0), (1, 0), -xi, xi + 1)]
+ ....: ((1, -1), (0, -1), 1/3*xi + 2/3, -1/3*xi + 1/3),
+ ....: ((0, 1), (4, 0), xi + 2, -xi - 1),
+ ....: ((2, 0), (5, 1), xi, -xi + 1),
+ ....: ((1, 0), (5, 0), xi + 1, -xi)]
sage: eq_up_to_order(sols, expected)
True
"""
@@ -2715,10 +2715,10 @@ def solve_S_unit_equation(K, S, prec=106, include_exponents=True, include_bound=
sage: S = K.primes_above(3)
sage: sols = solve_S_unit_equation(K, S, 200)
sage: expected = [
- ....: ((2, 1), (4, 0), xi + 2, -xi - 1),
- ....: ((5, -1), (4, -1), 1/3*xi + 2/3, -1/3*xi + 1/3),
- ....: ((5, 0), (1, 0), -xi, xi + 1),
- ....: ((1, 1), (2, 0), -xi + 1, xi)]
+ ....: ((0, 1), (4, 0), xi + 2, -xi - 1),
+ ....: ((1, -1), (0, -1), 1/3*xi + 2/3, -1/3*xi + 1/3),
+ ....: ((1, 0), (5, 0), xi + 1, -xi),
+ ....: ((2, 0), (5, 1), xi, -xi + 1)]
sage: eq_up_to_order(sols, expected)
True
@@ -2726,7 +2726,7 @@ def solve_S_unit_equation(K, S, prec=106, include_exponents=True, include_bound=
sage: solutions, bound = solve_S_unit_equation(K, S, 100, include_bound=True)
sage: bound
- 6
+ 7
You can omit the exponent vectors::
diff --git a/src/sage/rings/number_field/class_group.py b/src/sage/rings/number_field/class_group.py
index 46d0ca8c9d..1ad6d583a8 100644
--- a/src/sage/rings/number_field/class_group.py
+++ b/src/sage/rings/number_field/class_group.py
@@ -157,7 +157,7 @@ class FractionalIdealClass(AbelianGroupWithValuesElement):
sage: C=K.class_group()
sage: c = C(2, a)
sage: c^2
- Fractional ideal class (2, a^2 + 2*a - 1)
+ Fractional ideal class (4, a)
sage: c^3
Trivial principal fractional ideal class
sage: c^1000
@@ -467,7 +467,7 @@ class ClassGroup(AbelianGroupWithValues_class):
sage: CK = K.class_group()
sage: CL = L.class_group()
sage: [CL(I).exponents() for I in CK]
- [(0,), (4,), (2,)]
+ [(0,), (2,), (4,)]
"""
if isinstance(args[0], FractionalIdealClass):
return self.element_class(self, None, self._number_field.ideal(args[0].ideal()))
diff --git a/src/sage/rings/number_field/number_field.py b/src/sage/rings/number_field/number_field.py
index b0fbfffb96..937f7f533d 100644
--- a/src/sage/rings/number_field/number_field.py
+++ b/src/sage/rings/number_field/number_field.py
@@ -3513,7 +3513,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
sage: L. = K.extension(x^2 - 3, x^2 + 1)
sage: M. = L.extension(x^2 + 1)
sage: L.ideal(K.ideal(2, a))
- Fractional ideal (a)
+ Fractional ideal (-a)
sage: M.ideal(K.ideal(2, a)) == M.ideal(a*(b - c)/2)
True
@@ -4681,7 +4681,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
1/13*a^2 + 7/13*a - 332/13,
-1/13*a^2 + 6/13*a + 345/13,
-1,
- 2/13*a^2 + 1/13*a - 755/13]
+ -2/13*a^2 - 1/13*a + 755/13]
sage: units[5] in (1/13*a^2 - 19/13*a - 7/13, 1/13*a^2 + 20/13*a - 7/13)
True
sage: len(units) == 6
@@ -4723,8 +4723,8 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
sage: K. = QuadraticField(-105)
sage: K._S_class_group_quotient_matrix((K.ideal(11, a + 4),))
[0 0]
- [1 0]
[0 1]
+ [1 0]
"""
from sage.matrix.constructor import matrix
S_clgp_gens = self._S_class_group_and_units(S)[1]
@@ -4845,7 +4845,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
1/13*a^2 + 7/13*a - 332/13,
-1/13*a^2 + 6/13*a + 345/13,
-1,
- 2/13*a^2 + 1/13*a - 755/13]
+ -2/13*a^2 - 1/13*a + 755/13]
sage: gens[5] in (1/13*a^2 - 19/13*a - 7/13, 1/13*a^2 + 20/13*a - 7/13)
True
sage: gens[6] in (-1/13*a^2 + 45/13*a - 97/13, 1/13*a^2 - 45/13*a + 97/13)
@@ -6022,28 +6022,37 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
try:
return self._integral_basis_dict[v]
except (AttributeError, KeyError):
- f = self.pari_polynomial("y")
- if v:
- B = f.nfbasis(fa=v)
- elif self._assume_disc_small:
- B = f.nfbasis(1)
- elif not important:
- # Trial divide the discriminant with primes up to 10^6
- m = self.pari_polynomial().poldisc().abs().factor(limit=10**6)
- # Since we only need a *squarefree* factorization for
- # primes with exponent 1, we need trial division up to D^(1/3)
- # instead of D^(1/2).
- trialdivlimit2 = pari(10**12)
- trialdivlimit3 = pari(10**18)
- if all(p < trialdivlimit2 or (e == 1 and p < trialdivlimit3) or p.isprime() for p, e in zip(m[0], m[1])):
- B = f.nfbasis(fa = m)
- else:
- raise RuntimeError("Unable to factor discriminant with trial division")
+ pass
+
+ f = self.pari_polynomial("y")
+ if v:
+ # NOTE: here we make pari know about potentially big primes factors of
+ # the discriminant, see
+ # https://pari.math.u-bordeaux.fr/cgi-bin/bugreport.cgi?bug=2257
+ primelimit = pari.default("primelimit")
+ primes = [p for p in v if p > primelimit]
+ if primes:
+ pari.addprimes(primes)
+ B = f.nfbasis(fa=v)
+ elif self._assume_disc_small:
+ B = f.nfbasis(1)
+ elif not important:
+ # Trial divide the discriminant with primes up to 10^6
+ m = self.pari_polynomial().poldisc().abs().factor(limit=10**6)
+ # Since we only need a *squarefree* factorization for
+ # primes with exponent 1, we need trial division up to D^(1/3)
+ # instead of D^(1/2).
+ trialdivlimit2 = pari(10**12)
+ trialdivlimit3 = pari(10**18)
+ if all(p < trialdivlimit2 or (e == 1 and p < trialdivlimit3) or p.isprime() for p, e in zip(m[0], m[1])):
+ B = f.nfbasis(fa = m)
else:
- B = f.nfbasis()
+ raise RuntimeError("Unable to factor discriminant with trial division")
+ else:
+ B = f.nfbasis()
- self._integral_basis_dict[v] = B
- return B
+ self._integral_basis_dict[v] = B
+ return B
def reduced_basis(self, prec=None):
r"""
@@ -6729,7 +6738,8 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
"""
Return generators for the unit group modulo torsion.
- ALGORITHM: Uses PARI's :pari:`bnfunit` command.
+ ALGORITHM: Uses PARI's :pari:`bnfinit` command (that computes fundamental units
+ among other things).
INPUT:
@@ -6751,7 +6761,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
sage: A = x^4 - 10*x^3 + 20*5*x^2 - 15*5^2*x + 11*5^3
sage: K = NumberField(A, 'a')
sage: K.units()
- (8/275*a^3 - 12/55*a^2 + 15/11*a - 3,)
+ (-1/275*a^3 - 4/55*a^2 + 5/11*a - 3,)
For big number fields, provably computing the unit group can
take a very long time. In this case, one can ask for the
@@ -6762,14 +6772,14 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
sage: K.units(proof=True) # takes forever, not tested
...
sage: K.units(proof=False) # result not independently verified
- (a^9 + a - 1,
- a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2 - 2*a + 2,
- a^16 - a^15 + a^14 - a^12 + a^11 - a^10 - a^8 + a^7 - 2*a^6 + a^4 - 3*a^3 + 2*a^2 - 2*a + 1,
+ (-a^9 - a + 1,
+ -a^16 + a^15 - a^14 + a^12 - a^11 + a^10 + a^8 - a^7 + 2*a^6 - a^4 + 3*a^3 - 2*a^2 + 2*a - 1,
2*a^16 - a^14 - a^13 + 3*a^12 - 2*a^10 + a^9 + 3*a^8 - 3*a^6 + 3*a^5 + 3*a^4 - 2*a^3 - 2*a^2 + 3*a + 4,
- 2*a^16 - 3*a^15 + 3*a^14 - 3*a^13 + 3*a^12 - a^11 + a^9 - 3*a^8 + 4*a^7 - 5*a^6 + 6*a^5 - 4*a^4 + 3*a^3 - 2*a^2 - 2*a + 4,
- a^16 - a^15 - 3*a^14 - 4*a^13 - 4*a^12 - 3*a^11 - a^10 + 2*a^9 + 4*a^8 + 5*a^7 + 4*a^6 + 2*a^5 - 2*a^4 - 6*a^3 - 9*a^2 - 9*a - 7,
a^15 + a^14 + 2*a^11 + a^10 - a^9 + a^8 + 2*a^7 - a^5 + 2*a^3 - a^2 - 3*a + 1,
- 5*a^16 - 6*a^14 + a^13 + 7*a^12 - 2*a^11 - 7*a^10 + 4*a^9 + 7*a^8 - 6*a^7 - 7*a^6 + 8*a^5 + 6*a^4 - 11*a^3 - 5*a^2 + 13*a + 4)
+ -a^16 - a^15 - a^14 - a^13 - a^12 - a^11 - a^10 - a^9 - a^8 - a^7 - a^6 - a^5 - a^4 - a^3 - a^2 + 2,
+ -2*a^16 + 3*a^15 - 3*a^14 + 3*a^13 - 3*a^12 + a^11 - a^9 + 3*a^8 - 4*a^7 + 5*a^6 - 6*a^5 + 4*a^4 - 3*a^3 + 2*a^2 + 2*a - 4,
+ a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2 - 2*a + 2,
+ -a^14 - a^13 + a^12 + 2*a^10 + a^8 - 2*a^7 - 2*a^6 + 2*a^3 - a^2 + 2*a - 2)
TESTS:
@@ -6778,7 +6788,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
sage: K. = NumberField(1/2*x^2 - 1/6)
sage: K.units()
- (3*a - 2,)
+ (-3*a + 2,)
"""
proof = proof_flag(proof)
@@ -6796,7 +6806,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
pass
# get PARI to compute the units
- B = self.pari_bnf(proof).bnfunit()
+ B = self.pari_bnf(proof).bnf_get_fu()
B = tuple(self(b, check=False) for b in B)
if proof:
# cache the provable results and return them
@@ -6811,7 +6821,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
"""
Return the unit group (including torsion) of this number field.
- ALGORITHM: Uses PARI's :pari:`bnfunit` command.
+ ALGORITHM: Uses PARI's :pari:`bnfinit` and :pari:`bnfunits`.
INPUT:
@@ -6857,7 +6867,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
sage: U.gens()
(u0, u1, u2, u3, u4, u5, u6, u7, u8)
sage: U.gens_values() # result not independently verified
- [-1, a^9 + a - 1, a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2 - 2*a + 2, a^16 - a^15 + a^14 - a^12 + a^11 - a^10 - a^8 + a^7 - 2*a^6 + a^4 - 3*a^3 + 2*a^2 - 2*a + 1, 2*a^16 - a^14 - a^13 + 3*a^12 - 2*a^10 + a^9 + 3*a^8 - 3*a^6 + 3*a^5 + 3*a^4 - 2*a^3 - 2*a^2 + 3*a + 4, 2*a^16 - 3*a^15 + 3*a^14 - 3*a^13 + 3*a^12 - a^11 + a^9 - 3*a^8 + 4*a^7 - 5*a^6 + 6*a^5 - 4*a^4 + 3*a^3 - 2*a^2 - 2*a + 4, a^16 - a^15 - 3*a^14 - 4*a^13 - 4*a^12 - 3*a^11 - a^10 + 2*a^9 + 4*a^8 + 5*a^7 + 4*a^6 + 2*a^5 - 2*a^4 - 6*a^3 - 9*a^2 - 9*a - 7, a^15 + a^14 + 2*a^11 + a^10 - a^9 + a^8 + 2*a^7 - a^5 + 2*a^3 - a^2 - 3*a + 1, 5*a^16 - 6*a^14 + a^13 + 7*a^12 - 2*a^11 - 7*a^10 + 4*a^9 + 7*a^8 - 6*a^7 - 7*a^6 + 8*a^5 + 6*a^4 - 11*a^3 - 5*a^2 + 13*a + 4]
+ [-1, -a^9 - a + 1, -a^16 + a^15 - a^14 + a^12 - a^11 + a^10 + a^8 - a^7 + 2*a^6 - a^4 + 3*a^3 - 2*a^2 + 2*a - 1, 2*a^16 - a^14 - a^13 + 3*a^12 - 2*a^10 + a^9 + 3*a^8 - 3*a^6 + 3*a^5 + 3*a^4 - 2*a^3 - 2*a^2 + 3*a + 4, a^15 + a^14 + 2*a^11 + a^10 - a^9 + a^8 + 2*a^7 - a^5 + 2*a^3 - a^2 - 3*a + 1, -a^16 - a^15 - a^14 - a^13 - a^12 - a^11 - a^10 - a^9 - a^8 - a^7 - a^6 - a^5 - a^4 - a^3 - a^2 + 2, -2*a^16 + 3*a^15 - 3*a^14 + 3*a^13 - 3*a^12 + a^11 - a^9 + 3*a^8 - 4*a^7 + 5*a^6 - 6*a^5 + 4*a^4 - 3*a^3 + 2*a^2 + 2*a - 4, a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2 - 2*a + 2, -a^14 - a^13 + a^12 + 2*a^10 + a^8 - 2*a^7 - 2*a^6 + 2*a^3 - a^2 + 2*a - 2]
"""
proof = proof_flag(proof)
@@ -7045,7 +7055,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
sage: solutions, bound = K.S_unit_solutions(S, prec=100, include_bound=True)
sage: bound
- 6
+ 7
"""
from .S_unit_solver import solve_S_unit_equation
return solve_S_unit_equation(self, S, prec, include_exponents, include_bound, proof)
@@ -10884,9 +10894,9 @@ class NumberField_cyclotomic(NumberField_absolute):
EXAMPLES::
sage: k5. = CyclotomicField(5)
- sage: gap('E(5)^7 + 3')
+ sage: w = libgap.eval('E(5)^7 + 3')
+ sage: w
-3*E(5)-2*E(5)^2-3*E(5)^3-3*E(5)^4
- sage: w = gap('E(5)^7 + 3')
sage: z^7 + 3
z^2 + 3
sage: k5(w) # indirect doctest
@@ -10897,7 +10907,7 @@ class NumberField_cyclotomic(NumberField_absolute):
sage: F = CyclotomicField(8)
sage: z = F.gen()
- sage: a = gap(z+1/z); a
+ sage: a = libgap(z+1/z); a
E(8)-E(8)^3
sage: F(a)
-zeta8^3 + zeta8
@@ -10911,6 +10921,7 @@ class NumberField_cyclotomic(NumberField_absolute):
It also works with the old pexpect interface to GAP::
+ sage: a = gap(z + 1/z)
sage: b = gap(Matrix(F,[[z^2,1],[0,a+1]])); b
[ [ E(4), 1 ], [ 0, 1+E(8)-E(8)^3 ] ]
sage: b[1,2]
diff --git a/src/sage/rings/number_field/number_field_element.pyx b/src/sage/rings/number_field/number_field_element.pyx
index 0c2499bfee..88bfc6d6d8 100644
--- a/src/sage/rings/number_field/number_field_element.pyx
+++ b/src/sage/rings/number_field/number_field_element.pyx
@@ -1629,7 +1629,7 @@ cdef class NumberFieldElement(FieldElement):
sage: Q. = K[]
sage: L. = NumberField(X^4 + a)
sage: t = (-a).is_norm(L, element=True); t
- (True, b^3 + 1)
+ (True, -b^3 - 1)
sage: t[1].norm(K)
-a
@@ -1744,11 +1744,11 @@ cdef class NumberFieldElement(FieldElement):
sage: Q. = K[]
sage: L. = NumberField(X^4 + a)
sage: t = (-a)._rnfisnorm(L); t
- (b^3 + 1, 1)
+ (-b^3 - 1, 1)
sage: t[0].norm(K)
-a
sage: t = K(3)._rnfisnorm(L); t
- (-b^3 - a*b^2 - a^2*b + 1, 3*a^2 - 3*a + 6)
+ (b^3 + a*b^2 + a^2*b - 1, 3*a^2 - 3*a + 6)
sage: t[0].norm(K)*t[1]
3
@@ -1801,7 +1801,7 @@ cdef class NumberFieldElement(FieldElement):
raise ValueError("L (=%s) must be a relative number field with base field K (=%s) in rnfisnorm" % (L, K))
rnf_data = K.pari_rnfnorm_data(L, proof=proof)
- x, q = self.__pari__().rnfisnorm(rnf_data)
+ x, q = pari.rnfisnorm(rnf_data, self)
return L(x, check=False), K(q, check=False)
def _mpfr_(self, R):
diff --git a/src/sage/rings/number_field/number_field_ideal.py b/src/sage/rings/number_field/number_field_ideal.py
index 2e6a368843..aa2cc3c297 100644
--- a/src/sage/rings/number_field/number_field_ideal.py
+++ b/src/sage/rings/number_field/number_field_ideal.py
@@ -231,7 +231,7 @@ class NumberFieldIdeal(Ideal_generic):
sage: K. = NumberField(x^2 + 3); K
Number Field in a with defining polynomial x^2 + 3
sage: f = K.factor(15); f
- (Fractional ideal (-a))^2 * (Fractional ideal (5))
+ (Fractional ideal (1/2*a + 3/2))^2 * (Fractional ideal (5))
sage: (f[0][0] < f[1][0])
True
sage: (f[0][0] == f[0][0])
@@ -620,7 +620,7 @@ class NumberFieldIdeal(Ideal_generic):
sage: K. = CyclotomicField(7)
sage: I = K.factor(11)[0][0]; I
- Fractional ideal (-2*z^4 - 2*z^2 - 2*z + 1)
+ Fractional ideal (-3*z^4 - 2*z^3 - 2*z^2 - 2)
sage: A = I.free_module()
sage: A # warning -- choice of basis can be somewhat random
Free module of degree 6 and rank 6 over Integer Ring
@@ -3118,7 +3118,7 @@ class NumberFieldFractionalIdeal(MultiplicativeGroupElement, NumberFieldIdeal):
sage: K. = NumberField(x^5 + 2); K
Number Field in a with defining polynomial x^5 + 2
sage: f = K.factor(19); f
- (Fractional ideal (a^2 + a - 3)) * (Fractional ideal (-2*a^4 - a^2 + 2*a - 1)) * (Fractional ideal (a^2 + a - 1))
+ (Fractional ideal (a^2 + a - 3)) * (Fractional ideal (2*a^4 + a^2 - 2*a + 1)) * (Fractional ideal (a^2 + a - 1))
sage: [i.residue_class_degree() for i, _ in f]
[2, 2, 1]
"""
diff --git a/src/sage/rings/number_field/number_field_ideal_rel.py b/src/sage/rings/number_field/number_field_ideal_rel.py
index 37c20150ad..078682fa1f 100644
--- a/src/sage/rings/number_field/number_field_ideal_rel.py
+++ b/src/sage/rings/number_field/number_field_ideal_rel.py
@@ -18,7 +18,7 @@ EXAMPLES::
sage: G = [from_A(z) for z in I.gens()]; G
[7, -2*b*a - 1]
sage: K.fractional_ideal(G)
- Fractional ideal (2*b*a + 1)
+ Fractional ideal ((1/2*b + 2)*a - 1/2*b + 2)
sage: K.fractional_ideal(G).absolute_norm().factor()
7^2
"""
diff --git a/src/sage/rings/number_field/number_field_rel.py b/src/sage/rings/number_field/number_field_rel.py
index 312cbdf874..436af50e87 100644
--- a/src/sage/rings/number_field/number_field_rel.py
+++ b/src/sage/rings/number_field/number_field_rel.py
@@ -396,18 +396,18 @@ class NumberField_relative(NumberField_generic):
sage: K. = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: K.subfields(2)
[
- (Number Field in c0 with defining polynomial x^2 - 24*x + 72, Ring morphism:
- From: Number Field in c0 with defining polynomial x^2 - 24*x + 72
+ (Number Field in c0 with defining polynomial x^2 - 24*x + 96, Ring morphism:
+ From: Number Field in c0 with defining polynomial x^2 - 24*x + 96
To: Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
- Defn: c0 |--> -6*a + 12, None),
+ Defn: c0 |--> -4*b + 12, None),
(Number Field in c1 with defining polynomial x^2 - 24*x + 120, Ring morphism:
From: Number Field in c1 with defining polynomial x^2 - 24*x + 120
To: Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
Defn: c1 |--> 2*b*a + 12, None),
- (Number Field in c2 with defining polynomial x^2 - 24*x + 96, Ring morphism:
- From: Number Field in c2 with defining polynomial x^2 - 24*x + 96
+ (Number Field in c2 with defining polynomial x^2 - 24*x + 72, Ring morphism:
+ From: Number Field in c2 with defining polynomial x^2 - 24*x + 72
To: Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
- Defn: c2 |--> -4*b + 12, None)
+ Defn: c2 |--> -6*a + 12, None)
]
sage: K.subfields(8, 'w')
[
diff --git a/src/sage/rings/number_field/order.py b/src/sage/rings/number_field/order.py
index b5cdeb2a96..be1548f7a2 100644
--- a/src/sage/rings/number_field/order.py
+++ b/src/sage/rings/number_field/order.py
@@ -2180,7 +2180,7 @@ def EisensteinIntegers(names="omega"):
sage: R
Eisenstein Integers in Number Field in omega with defining polynomial x^2 + x + 1 with omega = -0.50000000000000000? + 0.866025403784439?*I
sage: factor(3 + omega)
- (omega) * (-3*omega - 2)
+ (-1) * (-omega - 3)
sage: CC(omega)
-0.500000000000000 + 0.866025403784439*I
sage: omega.minpoly()
diff --git a/src/sage/rings/number_field/unit_group.py b/src/sage/rings/number_field/unit_group.py
index 5c94f7407d..2a3c617ca3 100644
--- a/src/sage/rings/number_field/unit_group.py
+++ b/src/sage/rings/number_field/unit_group.py
@@ -15,12 +15,12 @@ The first generator is a primitive root of unity in the field::
sage: UK.gens_values() # random
[-1/12*a^3 + 1/6*a, 1/24*a^3 + 1/4*a^2 - 1/12*a - 1]
sage: UK.gen(0).value()
- -1/12*a^3 + 1/6*a
+ 1/12*a^3 - 1/6*a
sage: UK.gen(0)
u0
sage: UK.gen(0) + K.one() # coerce abstract generator into number field
- -1/12*a^3 + 1/6*a + 1
+ 1/12*a^3 - 1/6*a + 1
sage: [u.multiplicative_order() for u in UK.gens()]
[4, +Infinity]
@@ -37,18 +37,18 @@ as elements of an abstract multiplicative group::
sage: UK(-1)
u0^2
sage: [UK(u) for u in (x^4-1).roots(K, multiplicities=False)]
- [1, u0^2, u0^3, u0]
+ [1, u0^2, u0, u0^3]
sage: UK.fundamental_units() # random
[1/24*a^3 + 1/4*a^2 - 1/12*a - 1]
sage: torsion_gen = UK.torsion_generator(); torsion_gen
u0
sage: torsion_gen.value()
- -1/12*a^3 + 1/6*a
+ 1/12*a^3 - 1/6*a
sage: UK.zeta_order()
4
sage: UK.roots_of_unity()
- [-1/12*a^3 + 1/6*a, -1, 1/12*a^3 - 1/6*a, 1]
+ [1/12*a^3 - 1/6*a, -1, -1/12*a^3 + 1/6*a, 1]
Exp and log functions provide maps between units as field elements and exponent
vectors with respect to the generators::
@@ -82,7 +82,7 @@ S-unit groups may be constructed, where S is a set of primes::
sage: SUK.rank()
4
sage: SUK.gens_values()
- [-1, a^2 + 1, a^5 + a^4 - a^2 - a - 1, a + 1, -a + 1]
+ [-1, a^2 + 1, -a^5 - a^4 + a^2 + a + 1, a + 1, a - 1]
sage: u = 9*prod(SUK.gens_values()); u
-18*a^5 - 18*a^4 - 18*a^3 - 9*a^2 + 9*a + 27
sage: SUK.log(u)
@@ -100,29 +100,29 @@ A relative number field example::
sage: UL.zeta_order()
24
sage: UL.roots_of_unity()
- [-b*a - b,
- b^2*a,
- b^3,
- a + 1,
- -b*a,
- -b^2,
- b^3*a + b^3,
- a,
- b,
+ [-b*a,
-b^2*a - b^2,
- b^3*a,
- -1,
- b*a + b,
- -b^2*a,
-b^3,
- -a - 1,
- b*a,
- b^2,
- -b^3*a - b^3,
-a,
+ -b*a - b,
+ -b^2,
+ b^3*a,
+ -a - 1,
-b,
+ b^2*a,
+ b^3*a + b^3,
+ -1,
+ b*a,
b^2*a + b^2,
+ b^3,
+ a,
+ b*a + b,
+ b^2,
-b^3*a,
+ a + 1,
+ b,
+ -b^2*a,
+ -b^3*a - b^3,
1]
A relative extension example, which worked thanks to the code review by F.W.Clarke::
@@ -199,7 +199,7 @@ class UnitGroup(AbelianGroupWithValues_class):
sage: UK.gen(5)
u5
sage: UK.gen(5).value()
- z^7 + z
+ -z^7 - z
An S-unit group::
@@ -216,7 +216,7 @@ class UnitGroup(AbelianGroupWithValues_class):
sage: SUK.zeta_order()
26
sage: SUK.log(21*z)
- (12, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1)
+ (25, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1)
"""
# This structure is not a parent in the usual sense. The
# "elements" are NumberFieldElement_absolute. Instead, they should
@@ -250,7 +250,7 @@ class UnitGroup(AbelianGroupWithValues_class):
sage: UK.gens()
(u0, u1)
sage: UK.gens_values()
- [-1, 6*a - 37]
+ [-1, -6*a + 37]
sage: K. = QuadraticField(-3)
sage: UK = K.unit_group(); UK
@@ -258,7 +258,7 @@ class UnitGroup(AbelianGroupWithValues_class):
sage: UK.gens()
(u,)
sage: UK.gens_values()
- [1/2*a + 1/2]
+ [-1/2*a + 1/2]
sage: K. = CyclotomicField(13)
sage: UK = K.unit_group(); UK
@@ -323,34 +323,31 @@ class UnitGroup(AbelianGroupWithValues_class):
self.__S = S
self.__pS = pS = [P.pari_prime() for P in S]
- # compute the fundamental units via pari:
- fu = [K(u, check=False) for u in pK.bnfunit()]
- self.__nfu = len(fu)
-
- # compute the additional S-unit generators:
+ # compute units
+ # NOTE: old pari syntax for S-units (< 2.13.0): pK.bnfsunit(pS)
+ # NOTE: pari >= 2.13.0: the first component of the result of bnfunits
+ # are *all* units starting with S-units, followed by fundamental units
+ # followed by the torsion unit.
if S:
- self.__S_unit_data = pK.bnfsunit(pS)
- su = [K(u, check=False) for u in self.__S_unit_data[0]]
+ self.__S_unit_data = pK.bnfunits(pS)
else:
- su = []
- self.__nsu = len(su)
+ self.__S_unit_data = pK.bnfunits()
+ # TODO: converting the factored matrix representation of bnfunits into polynomial
+ # form is a *big* waste of time
+ su_fu_tu = [pK.nfbasistoalg(pK.nffactorback(z)) for z in self.__S_unit_data[0]]
+
+ self.__nfu = len(pK.bnf_get_fu()) # number of fundamental units
+ self.__nsu = len(su_fu_tu) - self.__nfu - 1 # number of S-units
+ self.__ntu = pK.bnf_get_tu()[0] # order of torsion
self.__rank = self.__nfu + self.__nsu
- # compute a torsion generator and pick the 'simplest' one:
- n, z = pK[7][3] # number of roots of unity and bnf.tu as in pari documentation
- n = ZZ(n)
- self.__ntu = n
- z = K(z, check=False)
-
- # If we replaced z by another torsion generator we would need
- # to allow for this in the dlog function! So we do not.
+ # Move the torsion unit first, then fundamental units then S-units
+ gens = [K(u, check=False) for u in su_fu_tu]
+ gens = [gens[-1]] + gens[self.__nsu:-1] + gens[:self.__nsu]
- # Store the actual generators (torsion first):
- gens = [z] + fu + su
- values = Sequence(gens, immutable=True, universe=self, check=False)
# Construct the abtract group:
- gens_orders = tuple([ZZ(n)]+[ZZ(0)]*(self.__rank))
- AbelianGroupWithValues_class.__init__(self, gens_orders, 'u', values, number_field)
+ gens_orders = tuple([ZZ(self.__ntu)]+[ZZ(0)]*(self.__rank))
+ AbelianGroupWithValues_class.__init__(self, gens_orders, 'u', gens, number_field)
def _element_constructor_(self, u):
"""
@@ -375,7 +372,7 @@ class UnitGroup(AbelianGroupWithValues_class):
sage: UK.gens()
(u0, u1)
sage: UK.gens_values()
- [-1, 6*a - 37]
+ [-1, -6*a + 37]
sage: UK.ngens()
2
sage: [UK(u) for u in UK.gens()]
@@ -394,8 +391,8 @@ class UnitGroup(AbelianGroupWithValues_class):
except TypeError:
raise ValueError("%s is not an element of %s"%(u,K))
if self.__S:
- m = pK.bnfissunit(self.__S_unit_data, pari(u)).mattranspose()
- if m.ncols()==0:
+ m = pK.bnfisunit(pari(u), self.__S_unit_data).mattranspose()
+ if m.ncols() == 0:
raise ValueError("%s is not an S-unit"%u)
else:
if not u.is_integral() or u.norm().abs() != 1:
@@ -405,9 +402,8 @@ class UnitGroup(AbelianGroupWithValues_class):
# convert column matrix to a list:
m = [ZZ(m[0,i].sage()) for i in range(m.ncols())]
- # NB pari puts the torsion after the fundamental units, before
- # the extra S-units but we have the torsion first:
- m = [m[self.__nfu]] + m[:self.__nfu] + m[self.__nfu+1:]
+ # NOTE: pari ordering for the units is (S-units, fundamental units, torsion unit)
+ m = [m[-1]] + m[self.__nsu:-1] + m[:self.__nsu]
return self.element_class(self, m)
@@ -527,9 +523,9 @@ class UnitGroup(AbelianGroupWithValues_class):
sage: U.zeta(2, all=True)
[-1]
sage: U.zeta(3)
- 1/2*z - 1/2
+ -1/2*z - 1/2
sage: U.zeta(3, all=True)
- [1/2*z - 1/2, -1/2*z - 1/2]
+ [-1/2*z - 1/2, 1/2*z - 1/2]
sage: U.zeta(4)
Traceback (most recent call last):
...
@@ -645,7 +641,7 @@ class UnitGroup(AbelianGroupWithValues_class):
sage: SUK = UnitGroup(K,S=2)
sage: v = (3,1,4,1,5,9,2)
sage: u = SUK.exp(v); u
- -8732*z^11 + 15496*z^10 + 51840*z^9 + 68804*z^8 + 51840*z^7 + 15496*z^6 - 8732*z^5 + 34216*z^3 + 64312*z^2 + 64312*z + 34216
+ 8732*z^11 - 15496*z^10 - 51840*z^9 - 68804*z^8 - 51840*z^7 - 15496*z^6 + 8732*z^5 - 34216*z^3 - 64312*z^2 - 64312*z - 34216
sage: SUK.log(u)
(3, 1, 4, 1, 5, 9, 2)
sage: SUK.log(u) == v
@@ -692,7 +688,7 @@ class UnitGroup(AbelianGroupWithValues_class):
sage: SUK = UnitGroup(K,S=2)
sage: v = (3,1,4,1,5,9,2)
sage: u = SUK.exp(v); u
- -8732*z^11 + 15496*z^10 + 51840*z^9 + 68804*z^8 + 51840*z^7 + 15496*z^6 - 8732*z^5 + 34216*z^3 + 64312*z^2 + 64312*z + 34216
+ 8732*z^11 - 15496*z^10 - 51840*z^9 - 68804*z^8 - 51840*z^7 - 15496*z^6 + 8732*z^5 - 34216*z^3 - 64312*z^2 - 64312*z - 34216
sage: SUK.log(u)
(3, 1, 4, 1, 5, 9, 2)
sage: SUK.log(u) == v
diff --git a/src/sage/rings/polynomial/multi_polynomial_ring.py b/src/sage/rings/polynomial/multi_polynomial_ring.py
index 077f380730..ce6e517c78 100644
--- a/src/sage/rings/polynomial/multi_polynomial_ring.py
+++ b/src/sage/rings/polynomial/multi_polynomial_ring.py
@@ -525,6 +525,8 @@ class MPolynomialRing_polydict( MPolynomialRing_macaulay2_repr, PolynomialRing_s
# univariate polynomials. Below, v is the variable
# with highest priority, and the x[i] are expressions
# in the remaining variables.
+ if x == 0:
+ return self.zero()
v = self.gens_dict_recursive()[str(x.variable())]
return sum(self(x[i]) * v ** i for i in range(x.poldegree() + 1))
diff --git a/src/sage/rings/polynomial/polynomial_element.pyx b/src/sage/rings/polynomial/polynomial_element.pyx
index b7f3058ce7..4615107773 100644
--- a/src/sage/rings/polynomial/polynomial_element.pyx
+++ b/src/sage/rings/polynomial/polynomial_element.pyx
@@ -7550,7 +7550,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
[(-3.5074662110434039?e451, 1)]
sage: p = bigc*x + 1
sage: p.roots(ring=RR)
- [(0.000000000000000, 1)]
+ [(-2.85106096489671e-452, 1)]
sage: p.roots(ring=AA)
[(-2.8510609648967059?e-452, 1)]
sage: p.roots(ring=QQbar)
diff --git a/src/sage/rings/polynomial/polynomial_quotient_ring.py b/src/sage/rings/polynomial/polynomial_quotient_ring.py
index 1ed1faaf15..6d4bb74464 100644
--- a/src/sage/rings/polynomial/polynomial_quotient_ring.py
+++ b/src/sage/rings/polynomial/polynomial_quotient_ring.py
@@ -1305,7 +1305,16 @@ class PolynomialQuotientRing_generic(CommutativeRing):
fixed in :trac:`14489`)::
sage: S.S_class_group([K.ideal(a)])
- [((1/4*xbar^2 + 31/4, (-1/8*a + 1/8)*xbar^2 - 31/8*a + 31/8, 1/16*xbar^3 + 1/16*xbar^2 + 31/16*xbar + 31/16, -1/16*a*xbar^3 + (1/16*a + 1/8)*xbar^2 - 31/16*a*xbar + 31/16*a + 31/8), 6), ((-1/4*xbar^2 - 23/4, (1/8*a - 1/8)*xbar^2 + 23/8*a - 23/8, -1/16*xbar^3 - 1/16*xbar^2 - 23/16*xbar - 23/16, 1/16*a*xbar^3 + (-1/16*a - 1/8)*xbar^2 + 23/16*a*xbar - 23/16*a - 23/8), 2)]
+ [((1/4*xbar^2 + 31/4,
+ (-1/8*a + 1/8)*xbar^2 - 31/8*a + 31/8,
+ 1/16*xbar^3 + 1/16*xbar^2 + 31/16*xbar + 31/16,
+ -1/16*a*xbar^3 + (1/16*a + 1/8)*xbar^2 - 31/16*a*xbar + 31/16*a + 31/8),
+ 6),
+ ((-1/4*xbar^2 - 23/4,
+ 1/4*a*xbar^2 + 23/4*a,
+ -1/8*xbar^3 + 1/8*xbar^2 - 23/8*xbar + 23/8,
+ (1/16*a - 1/16)*xbar^3 + (-1/16*a + 1/16)*xbar^2 + (23/16*a - 23/16)*xbar - 23/16*a + 23/16),
+ 2)]
Note that all the returned values live where we expect them to::
diff --git a/src/sage/schemes/elliptic_curves/ell_finite_field.py b/src/sage/schemes/elliptic_curves/ell_finite_field.py
index 87fbec3b69..adf0d2c73f 100644
--- a/src/sage/schemes/elliptic_curves/ell_finite_field.py
+++ b/src/sage/schemes/elliptic_curves/ell_finite_field.py
@@ -785,11 +785,11 @@ class EllipticCurve_finite_field(EllipticCurve_field, HyperellipticCurve_finite_
sage: len(E.gens())
2
sage: E.cardinality()
- 867361737988403547207212930746733987710588
+ 867361737988403547206134229616487867594472
sage: E.gens()[0].order()
- 433680868994201773603606465373366993855294
+ 433680868994201773603067114808243933797236
sage: E.gens()[1].order()
- 433680868994201773603606465373366993855294
+ 433680868994201773603067114808243933797236
"""
G = self.__pari__().ellgroup(flag=1)
return tuple(self.point(list(pt)) for pt in G[2])
diff --git a/src/sage/schemes/elliptic_curves/ell_generic.py b/src/sage/schemes/elliptic_curves/ell_generic.py
index ac5b4a98b4..194501bc94 100644
--- a/src/sage/schemes/elliptic_curves/ell_generic.py
+++ b/src/sage/schemes/elliptic_curves/ell_generic.py
@@ -2934,15 +2934,7 @@ class EllipticCurve_generic(WithEqualityById, plane_curve.ProjectivePlaneCurve):
sage: K. = QuadraticField(2)
sage: E = EllipticCurve([1,a])
sage: E.pari_curve()
- [Mod(0, y^2 - 2), Mod(0, y^2 - 2), Mod(0, y^2 - 2), Mod(1, y^2 - 2),
- Mod(y, y^2 - 2), Mod(0, y^2 - 2), Mod(2, y^2 - 2), Mod(4*y, y^2 - 2),
- Mod(-1, y^2 - 2), Mod(-48, y^2 - 2), Mod(-864*y, y^2 - 2),
- Mod(-928, y^2 - 2), Mod(3456/29, y^2 - 2), Vecsmall([5]),
- [[y^2 - 2, [2, 0], 8, 1, [[1, -1.41421356237310;
- 1, 1.41421356237310], [1, -1.41421356237310; 1, 1.41421356237310],
- [1, -1; 1, 1], [2, 0; 0, 4], [4, 0; 0, 2], [2, 0; 0, 1],
- [2, [0, 2; 1, 0]], []], [-1.41421356237310, 1.41421356237310],
- [1, y], [1, 0; 0, 1], [1, 0, 0, 2; 0, 1, 1, 0]]], [0, 0, 0, 0, 0]]
+ [Mod(0, y^2 - 2), Mod(0, y^2 - 2), Mod(0, y^2 - 2), Mod(1, y^2 - 2), Mod(y, y^2 - 2), Mod(0, y^2 - 2), Mod(2, y^2 - 2), Mod(4*y, y^2 - 2), Mod(-1, y^2 - 2), Mod(-48, y^2 - 2), Mod(-864*y, y^2 - 2), Mod(-928, y^2 - 2), Mod(3456/29, y^2 - 2), Vecsmall([5]), [[y^2 - 2, [2, 0], 8, 1, [[1, -1.41421356237310; 1, 1.41421356237310], [1, -1.41421356237310; 1, 1.41421356237310], [16, -23; 16, 23], [2, 0; 0, 4], [4, 0; 0, 2], [2, 0; 0, 1], [2, [0, 2; 1, 0]], [2]], [-1.41421356237310, 1.41421356237310], [1, y], [1, 0; 0, 1], [1, 0, 0, 2; 0, 1, 1, 0]]], [0, 0, 0, 0, 0]]
PARI no longer requires that the `j`-invariant has negative `p`-adic valuation::
diff --git a/src/sage/schemes/elliptic_curves/ell_number_field.py b/src/sage/schemes/elliptic_curves/ell_number_field.py
index dded480ba8..26f2d9c9d4 100644
--- a/src/sage/schemes/elliptic_curves/ell_number_field.py
+++ b/src/sage/schemes/elliptic_curves/ell_number_field.py
@@ -214,9 +214,9 @@ class EllipticCurve_number_field(EllipticCurve_field):
sage: E == loads(dumps(E))
True
sage: E.simon_two_descent()
- (2, 2, [(0 : 0 : 1), (1/8*a + 5/8 : -3/16*a - 7/16 : 1)])
+ (2, 2, [(0 : 0 : 1)])
sage: E.simon_two_descent(lim1=3, lim3=20, limtriv=5, maxprob=7, limbigprime=10)
- (2, 2, [(-1 : 0 : 1), (-1/8*a + 5/8 : -3/16*a - 9/16 : 1)])
+ (2, 2, [(-1 : 0 : 1), (5/32*a + 1/32 : -19/128*a + 41/128 : 1)])
::
@@ -244,22 +244,22 @@ class EllipticCurve_number_field(EllipticCurve_field):
C = Mod(y, y^2 + 7)
Computing L(S,2)
- L(S,2) = [Mod(Mod(-1/2*y + 1/2, y^2 + 7)*x^2 + Mod(-1/2*y - 1/2, y^2 + 7)*x + Mod(-y - 1, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(-x^2 + Mod(-1/2*y - 1/2, y^2 + 7)*x + 1, x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(-1, x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(x^2 + 2, x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(x + Mod(1/2*y + 3/2, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(x + Mod(1/2*y - 3/2, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))]
+ L(S,2) = [Mod(Mod(1/2*y - 1/2, y^2 + 7)*x^2 + Mod(1/2*y + 1/2, y^2 + 7)*x + Mod(y + 1, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(x^2 + Mod(1/2*y + 1/2, y^2 + 7)*x - 1, x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(-1, x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(x^2 + 2, x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(x + Mod(1/2*y + 3/2, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(x + Mod(1/2*y - 3/2, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))]
Computing the Selmer group
#LS2gen = 2
- LS2gen = [Mod(Mod(-1/2*y + 1/2, y^2 + 7)*x^2 + Mod(-1/2*y - 1/2, y^2 + 7)*x + Mod(-y - 1, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(x^2 + Mod(1/2*y + 1/2, y^2 + 7)*x - 1, x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))]
+ LS2gen = [Mod(x^2 + Mod(1/2*y + 1/2, y^2 + 7)*x - 1, x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(Mod(-1/2*y + 1/2, y^2 + 7)*x^2 + Mod(-1/2*y - 1/2, y^2 + 7)*x + Mod(-y - 1, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))]
Search for trivial points on the curve
Trivial points on the curve = [[Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7)], [1, 1, 0], [Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7), 1]]
+ zc = Mod(Mod(1, y^2 + 7)*x^2 + Mod(1/2*y + 1/2, y^2 + 7)*x + Mod(-1, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))
+ comes from the trivial point [Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7)]
zc = Mod(Mod(-1/2*y + 1/2, y^2 + 7)*x^2 + Mod(-1/2*y - 1/2, y^2 + 7)*x + Mod(-y - 1, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))
- Hilbert symbol (Mod(1, y^2 + 7),Mod(-2*y + 2, y^2 + 7)) =
+ Hilbert symbol (Mod(1, y^2 + 7),Mod(-2*y + 2, y^2 + 7)) =
sol of quadratic equation = [1, 1, 0]~
zc*z1^2 = Mod(4*x + Mod(-2*y + 6, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))
quartic: (-1)*Y^2 = x^4 + (3*y - 9)*x^2 + (-8*y + 16)*x + (9/2*y - 11/2)
reduced: Y^2 = -x^4 + (-3*y + 9)*x^2 + (-8*y + 16)*x + (-9/2*y + 11/2)
not ELS at [2, [0, 1]~, 1, 1, [1, -2; 1, 0]]
- zc = Mod(Mod(1, y^2 + 7)*x^2 + Mod(1/2*y + 1/2, y^2 + 7)*x + Mod(-1, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))
- comes from the trivial point [Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7)]
m1 = 1
m2 = 1
#S(E/K)[2] = 2
@@ -298,8 +298,8 @@ class EllipticCurve_number_field(EllipticCurve_field):
sage: E.simon_two_descent() # long time (4s on sage.math, 2013)
(3,
3,
- [(0 : 0 : 1),
- (-1/2*zeta43_0^2 - 1/2*zeta43_0 + 7 : -3/2*zeta43_0^2 - 5/2*zeta43_0 + 18 : 1)...)
+ [(5/8*zeta43_0^2 + 17/8*zeta43_0 - 9/4 : -27/16*zeta43_0^2 - 103/16*zeta43_0 + 39/8 : 1),
+ (0 : 0 : 1)])
"""
verbose = int(verbose)
if known_points is None:
diff --git a/src/sage/schemes/elliptic_curves/ell_rational_field.py b/src/sage/schemes/elliptic_curves/ell_rational_field.py
index 2953e80cac..b9d34cd64a 100644
--- a/src/sage/schemes/elliptic_curves/ell_rational_field.py
+++ b/src/sage/schemes/elliptic_curves/ell_rational_field.py
@@ -5407,9 +5407,9 @@ class EllipticCurve_rational_field(EllipticCurve_number_field):
sage: E = EllipticCurve('37a1')
sage: E.eval_modular_form([1.5+I,2.0+I,2.5+I],100) # abs tol 1e-20
- [-0.0018743978548152085771342944989052703431,
- 0.0018604485340371083710285594393397945456,
- -0.0018743978548152085771342944989052703431]
+ [-0.00187439785481520858 - 6.91083670607514589e-22*I,
+ 0.00186044853403710837 + 3.71914507780688601e-22*I,
+ -0.00187439785481520858 - 6.39417173217386647e-23*I]
sage: E.eval_modular_form(2.1+I, 100) # abs tol 1e-16
[0.00150864362757267079 + 0.00109100341113449845*I]
diff --git a/src/sage/schemes/elliptic_curves/isogeny_small_degree.py b/src/sage/schemes/elliptic_curves/isogeny_small_degree.py
index 7178da3685..ea0a7e88bc 100644
--- a/src/sage/schemes/elliptic_curves/isogeny_small_degree.py
+++ b/src/sage/schemes/elliptic_curves/isogeny_small_degree.py
@@ -801,8 +801,8 @@ def isogenies_5_0(E, minimal_models=True):
sage: K. = NumberField(x**6-320*x**3-320)
sage: E = EllipticCurve(K,[0,0,1,0,0])
sage: isogenies_5_0(E)
- [Isogeny of degree 5 from Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^6 - 320*x^3 - 320 to Elliptic Curve defined by y^2 + y = x^3 + (643/8*a^5-15779/48*a^4-32939/24*a^3-71989/2*a^2+214321/6*a-112115/3)*x + (2901961/96*a^5+4045805/48*a^4+12594215/18*a^3-30029635/6*a^2+15341626/3*a-38944312/9) over Number Field in a with defining polynomial x^6 - 320*x^3 - 320,
- Isogeny of degree 5 from Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^6 - 320*x^3 - 320 to Elliptic Curve defined by y^2 + y = x^3 + (-1109/8*a^5-53873/48*a^4-180281/24*a^3-14491/2*a^2+35899/6*a-43745/3)*x + (-17790679/96*a^5-60439571/48*a^4-77680504/9*a^3+1286245/6*a^2-4961854/3*a-73854632/9) over Number Field in a with defining polynomial x^6 - 320*x^3 - 320]
+ [Isogeny of degree 5 from Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^6 - 320*x^3 - 320 to Elliptic Curve defined by y^2 + y = x^3 + (241565/32*a^5-362149/48*a^4+180281/24*a^3-9693307/4*a^2+14524871/6*a-7254985/3)*x + (1660391123/192*a^5-829315373/96*a^4+77680504/9*a^3-66622345345/24*a^2+33276655441/12*a-24931615912/9) over Number Field in a with defining polynomial x^6 - 320*x^3 - 320,
+ Isogeny of degree 5 from Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^6 - 320*x^3 - 320 to Elliptic Curve defined by y^2 + y = x^3 + (47519/32*a^5-72103/48*a^4+32939/24*a^3-1909753/4*a^2+2861549/6*a-1429675/3)*x + (-131678717/192*a^5+65520419/96*a^4-12594215/18*a^3+5280985135/24*a^2-2637787519/12*a+1976130088/9) over Number Field in a with defining polynomial x^6 - 320*x^3 - 320]
"""
F = E.base_field()
if E.j_invariant() != 0:
diff --git a/src/sage/schemes/elliptic_curves/period_lattice.py b/src/sage/schemes/elliptic_curves/period_lattice.py
index 42793dc1b3..6ae1dd593d 100644
--- a/src/sage/schemes/elliptic_curves/period_lattice.py
+++ b/src/sage/schemes/elliptic_curves/period_lattice.py
@@ -1626,7 +1626,7 @@ class PeriodLattice_ell(PeriodLattice):
sage: embs = K.embeddings(CC)
sage: Lambda = E.period_lattice(embs[0])
sage: Lambda.elliptic_logarithm(P+3*Q, 100)
- 4.7100131126199672766973600998
+ 4.3543876242043418255250464574
sage: R. = QQ[]
sage: K. = NumberField(x^2 + x + 5)
sage: E = EllipticCurve(K, [0,0,1,-3,-5])
diff --git a/src/sage/schemes/toric/chow_group.py b/src/sage/schemes/toric/chow_group.py
index 89e82467b0..78730f930a 100644
--- a/src/sage/schemes/toric/chow_group.py
+++ b/src/sage/schemes/toric/chow_group.py
@@ -62,7 +62,7 @@ EXAMPLES::
7
sage: a = sum( A.gen(i) * (i+1) for i in range(A.ngens()) ) # an element of A
sage: a # long time (2s on sage.math, 2011)
- ( 3 | 1 mod 7 | 0 mod 2, 1 mod 2, 4, 5, 6, 7, 8 | 9 )
+ ( 3 | 1 mod 7 | 1 mod 2, 0 mod 2, 4, 5, 6, 7, 8 | 9 )
The Chow group elements are printed as ``( a0 | a1 mod 7 | a2 mod 2,
a3 mod 2, a4, a5, a6, a7, a8 | a9 )``, which denotes the element of
@@ -93,13 +93,13 @@ Cones of toric varieties can determine their own Chow cycle::
sage: cone = X.fan(dim=2)[3]; cone
2-d cone of Rational polyhedral fan in 3-d lattice N
sage: A_cone = A(cone); A_cone
- ( 0 | 1 mod 7 | 0 mod 2, 0 mod 2, 0, 0, 0, 0, 0 | 0 )
+ ( 0 | 6 mod 7 | 0 mod 2, 0 mod 2, 0, 0, 0, 0, 0 | 0 )
sage: A_cone.degree()
1
sage: 2 * A_cone
- ( 0 | 2 mod 7 | 0 mod 2, 0 mod 2, 0, 0, 0, 0, 0 | 0 )
+ ( 0 | 5 mod 7 | 0 mod 2, 0 mod 2, 0, 0, 0, 0, 0 | 0 )
sage: A_cone + A.gen(0)
- ( 0 | 1 mod 7 | 0 mod 2, 1 mod 2, 0, 0, 0, 0, 0 | 0 )
+ ( 0 | 6 mod 7 | 1 mod 2, 0 mod 2, 0, 0, 0, 0, 0 | 0 )
Chow cycles can be of mixed degrees::
diff --git a/src/sage/schemes/toric/homset.py b/src/sage/schemes/toric/homset.py
index 4bff92bcb0..ff6211cabe 100644
--- a/src/sage/schemes/toric/homset.py
+++ b/src/sage/schemes/toric/homset.py
@@ -467,12 +467,27 @@ class SchemeHomset_points_toric_field(SchemeHomset_points_toric_base):
sage: point_set.cardinality()
21
sage: sorted(X.point_set().list())
- [[0 : 0 : 1], [0 : 1 : 0], [0 : 1 : 1], [0 : 1 : 3],
- [1 : 0 : 0], [1 : 0 : 1], [1 : 0 : 3], [1 : 1 : 0],
- [1 : 1 : 1], [1 : 1 : 2], [1 : 1 : 3], [1 : 1 : 4],
- [1 : 1 : 5], [1 : 1 : 6], [1 : 3 : 0], [1 : 3 : 1],
- [1 : 3 : 2], [1 : 3 : 3], [1 : 3 : 4], [1 : 3 : 5],
- [1 : 3 : 6]]
+ [[0 : 0 : 1],
+ [0 : 1 : 0],
+ [0 : 1 : 1],
+ [0 : 1 : 5],
+ [1 : 0 : 0],
+ [1 : 0 : 1],
+ [1 : 0 : 5],
+ [1 : 1 : 0],
+ [1 : 1 : 1],
+ [1 : 1 : 2],
+ [1 : 1 : 3],
+ [1 : 1 : 4],
+ [1 : 1 : 5],
+ [1 : 1 : 6],
+ [1 : 3 : 1],
+ [1 : 3 : 2],
+ [1 : 3 : 3],
+ [1 : 3 : 4],
+ [1 : 3 : 5],
+ [1 : 3 : 6],
+ [1 : 5 : 0]]
As for a non-compact example, the blow-up of the plane is the line
bundle $O_{\mathbf{P}^1}(-1)$. Its point set is the Cartesian
@@ -641,7 +656,7 @@ class SchemeHomset_points_subscheme_toric_field(SchemeHomset_points_toric_base):
sage: P2. = toric_varieties.P2(base_ring=GF(5))
sage: cubic = P2.subscheme([x^3 + y^3 + z^3])
sage: list(cubic.point_set())
- [[0 : 1 : 4], [1 : 0 : 4], [1 : 4 : 0], [1 : 2 : 1], [1 : 1 : 2], [1 : 3 : 3]]
+ [[0 : 1 : 4], [1 : 0 : 4], [1 : 4 : 0], [1 : 1 : 2], [1 : 2 : 1], [1 : 3 : 3]]
sage: cubic.point_set().cardinality()
6
"""
@@ -661,7 +676,7 @@ class SchemeHomset_points_subscheme_toric_field(SchemeHomset_points_toric_base):
sage: P2. = toric_varieties.P2(base_ring=GF(5))
sage: cubic = P2.subscheme([x^3 + y^3 + z^3])
sage: list(cubic.point_set())
- [[0 : 1 : 4], [1 : 0 : 4], [1 : 4 : 0], [1 : 2 : 1], [1 : 1 : 2], [1 : 3 : 3]]
+ [[0 : 1 : 4], [1 : 0 : 4], [1 : 4 : 0], [1 : 1 : 2], [1 : 2 : 1], [1 : 3 : 3]]
sage: cubic.point_set().cardinality()
6
"""
diff --git a/src/sage/schemes/toric/points.py b/src/sage/schemes/toric/points.py
index 31e7769ede..361a010d2b 100644
--- a/src/sage/schemes/toric/points.py
+++ b/src/sage/schemes/toric/points.py
@@ -537,7 +537,7 @@ class FiniteFieldPointEnumerator(NaiveFinitePointEnumerator):
sage: enum._Chow_group_torsion()
((1, 2, 4), (1, 4, 2))
sage: enum._Chow_group_torsion_generators()
- ((1, 2, 4),)
+ ((1, 4, 2),)
"""
if self.fan.is_smooth():
return tuple()
@@ -673,7 +673,7 @@ class FiniteFieldPointEnumerator(NaiveFinitePointEnumerator):
sage: list(cokernel)
[(0), (1)]
sage: [p.lift() for p in cokernel]
- [(0, 0), (0, 1)]
+ [(0, 0), (0, -1)]
"""
from sage.matrix.constructor import matrix, block_matrix, identity_matrix
from sage.rings.all import ZZ
@@ -955,9 +955,9 @@ class FiniteFieldSubschemePointEnumerator(NaiveSubschemePointEnumerator):
sage: ffe.homogeneous_coordinates([0], nonzero_coordinates, cokernel)
(1, 1, 0)
sage: ffe.homogeneous_coordinates([1], nonzero_coordinates, cokernel)
- (1, 3, 0)
+ (1, 5, 0)
sage: ffe.homogeneous_coordinates([2], nonzero_coordinates, cokernel)
- (1, 2, 0)
+ (1, 4, 0)
"""
z = [self.ambient.ring.zero()] * len(self.ambient.rays())
z_nonzero = self.ambient.exp(
@@ -986,7 +986,7 @@ class FiniteFieldSubschemePointEnumerator(NaiveSubschemePointEnumerator):
sage: point_set = X.point_set()
sage: ffe = point_set._enumerator()
sage: list(ffe) # indirect doctest
- [(1, 4, 3), (1, 1, 6), (1, 2, 5)]
+ [(1, 1, 6), (1, 2, 5), (1, 4, 3)]
"""
for cone, nonzero_coordinates, cokernel in self.ambient.cone_points_iter():
R = PolynomialRing(self.ambient.ring, cokernel.ngens(), 't')
@@ -1011,11 +1011,11 @@ class FiniteFieldSubschemePointEnumerator(NaiveSubschemePointEnumerator):
sage: Y = X.subscheme(u^3 + v^3 + w^3 + u*v*w)
sage: point_set = Y.point_set()
sage: list(point_set)
- [[0 : 1 : 3],
- [1 : 0 : 3],
- [1 : 3 : 0],
- [1 : 1 : 6],
+ [[0 : 1 : 5],
+ [1 : 0 : 5],
+ [1 : 5 : 0],
[1 : 1 : 4],
+ [1 : 1 : 6],
[1 : 3 : 2],
[1 : 3 : 5]]
sage: ffe = point_set._enumerator()
diff --git a/src/sage/structure/factorization.py b/src/sage/structure/factorization.py
index 1d32db0842..7636f1a9ba 100644
--- a/src/sage/structure/factorization.py
+++ b/src/sage/structure/factorization.py
@@ -133,17 +133,17 @@ Factorizations can involve fairly abstract mathematical objects::
sage: K. = NumberField(x^2 + 3); K
Number Field in a with defining polynomial x^2 + 3
sage: f = K.factor(15); f
- (Fractional ideal (-a))^2 * (Fractional ideal (5))
+ (Fractional ideal (1/2*a + 3/2))^2 * (Fractional ideal (5))
sage: f.universe()
Monoid of ideals of Number Field in a with defining polynomial x^2 + 3
sage: f.unit()
Fractional ideal (1)
sage: g=K.factor(9); g
- (Fractional ideal (-a))^4
+ (Fractional ideal (1/2*a + 3/2))^4
sage: f.lcm(g)
- (Fractional ideal (-a))^4 * (Fractional ideal (5))
+ (Fractional ideal (1/2*a + 3/2))^4 * (Fractional ideal (5))
sage: f.gcd(g)
- (Fractional ideal (-a))^2
+ (Fractional ideal (1/2*a + 3/2))^2
sage: f.is_integral()
True
diff --git a/src/sage/tests/books/computational-mathematics-with-sagemath/linalg_doctest.py b/src/sage/tests/books/computational-mathematics-with-sagemath/linalg_doctest.py
index d67a33e7c4..7cba0ef6bb 100644
--- a/src/sage/tests/books/computational-mathematics-with-sagemath/linalg_doctest.py
+++ b/src/sage/tests/books/computational-mathematics-with-sagemath/linalg_doctest.py
@@ -232,13 +232,13 @@ Sage example in ./linalg.tex, line 1640::
sage: A = matrix(ZZ, 4, 5,\
....: [-1,-1,-1,-2,-2,-2,1,1,-1,2,2,2,2,2,-1,2,2,2,2,2])
sage: S,U,V = A.smith_form(); S,U,V
- (
- [ 0 -2 -1 -5 0]
- [1 0 0 0 0] [ 1 0 0 0] [ 1 0 1 -1 -1]
- [0 1 0 0 0] [ 0 0 1 0] [ 0 0 0 0 1]
- [0 0 3 0 0] [-2 1 0 0] [-1 2 0 5 0]
- [0 0 0 6 0], [ 0 0 -2 -1], [ 0 -1 0 -2 0]
- )
+ (
+ [ 0 -2 0 -5 0]
+ [1 0 0 0 0] [ 1 0 0 0] [ 1 0 0 -1 -1]
+ [0 1 0 0 0] [-2 1 1 0] [ 0 0 0 0 1]
+ [0 0 3 0 0] [-4 2 3 0] [-1 1 1 5 0]
+ [0 0 0 6 0], [ 4 -2 -2 -1], [ 0 0 -1 -2 0]
+ )
Sage example in ./linalg.tex, line 1674::
diff --git a/src/sage/tests/books/judson-abstract-algebra/galois-sage.py b/src/sage/tests/books/judson-abstract-algebra/galois-sage.py
index 6c25aa5dc5..8bdceb9424 100644
--- a/src/sage/tests/books/judson-abstract-algebra/galois-sage.py
+++ b/src/sage/tests/books/judson-abstract-algebra/galois-sage.py
@@ -385,69 +385,49 @@ r"""
sage: L.subfields()
[
- (Number Field in c0 with defining polynomial x,
- Ring morphism:
- From: Number Field in c0 with defining polynomial x
- To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
- Defn: 0 |--> 0,
- None),
- (Number Field in c1 with defining polynomial x^2 + 112*x + 40000,
- Ring morphism:
- From: Number Field in c1 with defining polynomial x^2 + 112*x + 40000
- To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
- Defn: c1 |--> 4*c^4,
- None),
- (Number Field in c2 with defining polynomial x^2 + 512,
- Ring morphism:
- From: Number Field in c2 with defining polynomial x^2 + 512
- To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
- Defn: c2 |--> 1/25*c^6 + 78/25*c^2,
- None),
- (Number Field in c3 with defining polynomial x^2 - 288,
- Ring morphism:
- From: Number Field in c3 with defining polynomial x^2 - 288
- To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
- Defn: c3 |--> -1/25*c^6 + 22/25*c^2,
- None),
- (Number Field in c4 with defining polynomial x^4 + 112*x^2 + 40000,
- Ring morphism:
- From: Number Field in c4 with defining polynomial x^4 + 112*x^2 + 40000
- To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
- Defn: c4 |--> 2*c^2,
- None),
- (Number Field in c5 with defining polynomial x^4 + 648,
- Ring morphism:
- From: Number Field in c5 with defining polynomial x^4 + 648
- To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
- Defn: c5 |--> 1/80*c^5 + 79/40*c,
- None),
- (Number Field in c6 with defining polynomial x^4 + 8,
- Ring morphism:
- From: Number Field in c6 with defining polynomial x^4 + 8
+ (Number Field in c0 with defining polynomial x, Ring morphism:
+ From: Number Field in c0 with defining polynomial x
To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
- Defn: c6 |--> -1/80*c^5 + 1/40*c,
- None),
- (Number Field in c7 with defining polynomial x^4 - 512,
- Ring morphism:
- From: Number Field in c7 with defining polynomial x^4 - 512
- To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
- Defn: c7 |--> -1/60*c^5 + 41/30*c,
- None),
- (Number Field in c8 with defining polynomial x^4 - 32,
- Ring morphism:
- From: Number Field in c8 with defining polynomial x^4 - 32
- To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
- Defn: c8 |--> 1/60*c^5 + 19/30*c,
- None),
- (Number Field in c9 with defining polynomial x^8 + 28*x^4 + 2500,
- Ring morphism:
- From: Number Field in c9 with defining polynomial x^8 + 28*x^4 + 2500
- To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
- Defn: c9 |--> c,
- Ring morphism:
- From: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
- To: Number Field in c9 with defining polynomial x^8 + 28*x^4 + 2500
- Defn: c |--> c9)
+ Defn: 0 |--> 0, None),
+ (Number Field in c1 with defining polynomial x^2 + 112*x + 40000, Ring morphism:
+ From: Number Field in c1 with defining polynomial x^2 + 112*x + 40000
+ To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
+ Defn: c1 |--> 4*c^4, None),
+ (Number Field in c2 with defining polynomial x^2 + 512, Ring morphism:
+ From: Number Field in c2 with defining polynomial x^2 + 512
+ To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
+ Defn: c2 |--> 1/25*c^6 + 78/25*c^2, None),
+ (Number Field in c3 with defining polynomial x^2 - 288, Ring morphism:
+ From: Number Field in c3 with defining polynomial x^2 - 288
+ To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
+ Defn: c3 |--> -1/25*c^6 + 22/25*c^2, None),
+ (Number Field in c4 with defining polynomial x^4 + 112*x^2 + 40000, Ring morphism:
+ From: Number Field in c4 with defining polynomial x^4 + 112*x^2 + 40000
+ To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
+ Defn: c4 |--> 2*c^2, None),
+ (Number Field in c5 with defining polynomial x^4 + 8, Ring morphism:
+ From: Number Field in c5 with defining polynomial x^4 + 8
+ To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
+ Defn: c5 |--> -1/80*c^5 + 1/40*c, None),
+ (Number Field in c6 with defining polynomial x^4 + 648, Ring morphism:
+ From: Number Field in c6 with defining polynomial x^4 + 648
+ To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
+ Defn: c6 |--> 1/80*c^5 + 79/40*c, None),
+ (Number Field in c7 with defining polynomial x^4 - 512, Ring morphism:
+ From: Number Field in c7 with defining polynomial x^4 - 512
+ To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
+ Defn: c7 |--> -1/60*c^5 + 41/30*c, None),
+ (Number Field in c8 with defining polynomial x^4 - 32, Ring morphism:
+ From: Number Field in c8 with defining polynomial x^4 - 32
+ To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
+ Defn: c8 |--> 1/60*c^5 + 19/30*c, None),
+ (Number Field in c9 with defining polynomial x^8 + 28*x^4 + 2500, Ring morphism:
+ From: Number Field in c9 with defining polynomial x^8 + 28*x^4 + 2500
+ To: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
+ Defn: c9 |--> c, Ring morphism:
+ From: Number Field in c with defining polynomial x^8 + 28*x^4 + 2500
+ To: Number Field in c9 with defining polynomial x^8 + 28*x^4 + 2500
+ Defn: c |--> c9)
]
~~~~~~~~~~~~~~~~~~~~~~ ::
diff --git a/src/sage/tests/parigp.py b/src/sage/tests/parigp.py
index c118b6eb37..f567ade058 100644
--- a/src/sage/tests/parigp.py
+++ b/src/sage/tests/parigp.py
@@ -2,15 +2,6 @@ r"""
This file is meant to catch errors in the PARI/GP package which are not
caught by any other tests.
-Check that :trac:`9876` has been fixed, this test comes from PARI's
-self-test :pari:`rnfkummer` but was modified such that the answer is
-canonical::
-
- sage: pari('addprimes([31438243, 238576291, 18775387483, 24217212463267, 1427657500359111961, 135564809928627323997297867381959])')
- [31438243, 238576291, 18775387483, 24217212463267, 1427657500359111961, 135564809928627323997297867381959]
- sage: pari('K = bnfinit(y^4-52*y^2+26,1); pol = rnfkummer(bnrinit(K,3,1),Mat(5)); L = rnfinit(K, pol); polredabs(polredbest(L.polabs))') # long time
- x^20 - 112*x^18 + 5108*x^16 - 123460*x^14 + 1724337*x^12 - 14266996*x^10 + 69192270*x^8 - 188583712*x^6 + 260329852*x^4 - 141461008*x^2 + 19860776
-
Check that :trac:`10195` (PARI bug 1153) has been fixed::
sage: print(gp.eval("mathnf([0,0,0,0,0,0,0,0,0,13;0,0,0,0,0,0,0,0,23,6;0,0,0,0,0,0,0,23,-4,-7;0,0,0,0,0,0,17,-3,5,-5;0,0,0,0,0,56,16,-16,-15,-17;0,0,0,0,57,24,-16,-25,2,-21;0,0,0,114,9,56,51,-52,25,-55;0,0,113,-31,-11,24,0,28,34,-16;0,50,3,2,16,-6,-2,7,-19,-21;118,43,51,23,37,-52,18,38,51,28],0)"))