id summary reporter owner description type status priority milestone component resolution keywords cc merged author reviewer upstream work_issues branch commit dependencies stopgaps
9409 Bug in elliptic curves method .count_points() over finite fields Adam Sorkin John Cremona "There is some bug in the method .count_points() which belongs to elliptic curves defined over finite fields. This might be specific to EC defined over number fields - I only get this error when I take an EC over a number field, reduce at a good prime and then count points. In fact, I get the correct answer the first time, but if I define a second EC over a possibly different number field and count points at a good reduction, then the method .count_points() fails. I suspect this has to do with the cacheing...
If you want to reproduce the behavior, try the following code:
{{{
### this just runs through the method outlined above:
def test(curve, bound):
for i in primes(bound):
print ""Checking primes over %d: ""%i
factors = curve.base_field().ideal(i).factor()
for j in range(len(factors)):
if curve.has_good_reduction(factors[j][0]):
if factors[j][0].divides(curve.discriminant()):
print ""Curve has good reduction, but this isn't not a minimal model"",
print ""at %s with %d points in the reduced curve""%(factors[j][0], curve.local_minimal_model(factors[j][0]).reduction(factors[j][0]).count_points() )
else:
print ""Curve has good reduction and is a minimal model""
print ""at %s with %d points in the reduced curve""%(factors[j][0], curve.reduction(factors[j][0]).count_points() )
else:
print ""Curve has bad reduction over %s""%factors[j][0]
return
### sample 1
K. = NumberField(x^2 + 1); E = EllipticCurve(K, [0, 1, 0, -2*t - 2, 2*t]); E
### sample 2
L.__ = NumberField(x^2 - 2); F = EllipticCurve(L, [0,2,0, 2*u +4, 2*u + 3]); F
test(E, 100)
## the above works fine; the next command will cause the error.
test(F, 100)
You will get the correct output for the first few primes, but the error message, which in the above case occurs above the prime ideal (67), is
Traceback (most recent call last):
File """", line 1, in
File ""_sage_input_8.py"", line 10, in
exec compile(u'open(""___code___.py"",""w"").write(""# -*- coding: utf-8 -*-\\n"" + _support_.preparse_worksheet_cell(base64.b64decode(""dGVzdChGLCAxMDAp""),globals())+""\\n""); execfile(os.path.abspath(""___code___.py""))
File """", line 1, in
File ""/tmp/tmpVYbgxh/___code___.py"", line 3, in
exec compile(u'test(F, _sage_const_100 )
File """", line 1, in
File ""/tmp/tmptawaYw/___code___.py"", line 14, in test
print ""at %s with %d points in the reduced curve""%(factors[j][_sage_const_0 ], curve.reduction(factors[j][_sage_const_0 ]).count_points() )
File ""/usr/local/sage2/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_finite_field.py"", line 322, in count_points
return self.cardinality()
File ""/usr/local/sage2/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_finite_field.py"", line 951, in cardinality
self._order = self.cardinality_bsgs()
File ""/usr/local/sage2/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_finite_field.py"", line 1220, in cardinality_bsgs
N1 = ZZ(2)**sum([e for P,e in E1._p_primary_torsion_basis(2)])
File ""/usr/local/sage2/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_generic.py"", line 2660, in _p_primary_torsion_basis
Ep = self(0).division_points(p)
File ""/usr/local/sage2/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_point.py"", line 879, in division_points
Q = E.lift_x(x)
File ""/usr/local/sage2/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_generic.py"", line 855, in lift_x
raise ValueError, ""No point with x-coordinate %s on %s""%(x, self)
ValueError: No point with x-coordinate 39*tbar + 11 on Elliptic Curve defined by y^2 = x^3 + 2*x^2 + (2*ubar+4)*x + (2*ubar+3) over Residue field in ubar of Fractional ideal (67)
}}}" defect closed major sage-duplicate/invalid/wontfix elliptic curves worksforme Elliptic Curves .count_points() finite fields N/A
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