I ran the following code in SageMath 10.2:
sage: K.<a> = QuadraticField(2)
sage: E = EllipticCurve([0, 60/49*a - 135/49, 0, -576/343*a + 904/343, 0]);
E
Elliptic Curve defined by y^2 = x^3 + (60/49*a-135/49)*x^2 +
(-576/343*a+904/343)*x over Number Field in a with defining polynomial x^2
- 2 with a = 1.414213562373095?
sage: E.rank()
1
So I know that the elliptic curve is properly defined, and there should be
a generator. But when I try to find the generator:
sage: E.gens_quadratic()
I get the following error:
---------------------------------------------------------------------------
TypeError Traceback (most recent call last) Cell In [5], line 1 ----> 1 E.
gens_quadratic() File
/ext/sage/10.2/src/sage/schemes/elliptic_curves/ell_number_field.py:4123,
in EllipticCurve_number_field.gens_quadratic(self, **kwds) 4121 gens1 =
[iso1(P) for P in EQ1.gens(**kwds)] 4122 gens2 = [iso2(P) for P in EQ2.gens(
**kwds)] -> 4123 gens = self.saturation(gens1 + gens2, max_prime=2)[0] 4124
self.__gens = gens 4125 return gens File
/ext/sage/10.2/src/sage/schemes/elliptic_curves/ell_number_field.py:4042,
in EllipticCurve_number_field.saturation(self, points, verbose, max_prime,
one_prime, odd_primes_only, lower_ht_bound, reg, debug) 4040 if verbose:
4041 print("Saturating at p=%s" % p) -> 4042 newPlist, expo = saturator.
full_p_saturation(Plist, p) 4043 if expo: 4044 if verbose: File
/ext/sage/10.2/src/sage/schemes/elliptic_curves/saturation.py:281, in
EllipticCurveSaturator.full_p_saturation(self, Plist, p) 278 if verbose: 279
print("Adding {} torsion generators before
{}-saturation".format(extra_torsion,p))
--> 281 res = self.p_saturation(Plist, p) 282 while res: # res is either
False or (i, newP) 283 exponent += 1 File
/ext/sage/10.2/src/sage/schemes/elliptic_curves/saturation.py:497, in
EllipticCurveSaturator.p_saturation(self, Plist, p, sieve) 495 cm_test =
E.has_rational_cm()
and kro(E.cm_discriminant(), p) == -1 496 for q in Primes(): --> 497 if any(
q.divides(m) for m in avoid): 498 continue 499 if cm_test and not p.
divides(q-1): File
/ext/sage/10.2/src/sage/schemes/elliptic_curves/saturation.py:497, in
<genexpr>(.0) 495 cm_test = E.has_rational_cm() and kro(E.cm_discriminant(),
p) == -1 496 for q in Primes(): --> 497 if any(q.divides(m) for m in
avoid): 498 continue 499 if cm_test and not p.divides(q-1): File
/ext/sage/10.2/src/sage/rings/integer.pyx:4171, in
sage.rings.integer.Integer.divides() 4169 cdef bint t 4170 cdef Integer _n ->
4171 _n = Integer(n) 4172 if mpz_sgn(self.value) == 0: 4173 return
mpz_sgn(_n.value) == 0 File /ext/sage/10.2/src/sage/rings/integer.pyx:653,
in sage.rings.integer.Integer.__init__() 651 otmp = getattr(x, "_integer_",
None) 652 if otmp is not None: --> 653 set_from_Integer(self,
otmp(the_integer_ring)) 654 return 655 File
/ext/sage/10.2/src/sage/rings/rational.pyx:2969, in
sage.rings.rational.Rational._integer_() 2967 """ 2968 if not
mpz_cmp_si(mpq_denref(self.value), 1) == 0: -> 2969 raise TypeError("no
conversion of this rational to integer") 2970 cdef Integer n =
Integer.__new__(Integer) 2971 n.set_from_mpz(mpq_numref(self.value))
TypeError: no conversion of this rational to integer One thing I noticed is
that this error is referring to Primes().
This field happens to have class number 1, but there are many quadratic
domains that have a larger class number, where Primes would be meaningless.
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