def dimension_cuspforms_sqrt5(k1, k2):
'''
Return the dimension of hilbert cusp forms of
weight (k1, k2) where k1 > 2 and k2 > 2.
cf. K. Takase, on the trace formula of the Hecke operators and the special
values of the second L-functions attached to the Hilbert modular forms.
manuscripta math. 55, 137 -- 170 (1986).
'''
k = (k1, k2)
F = QuadraticField(5)
rho = (1 + F.gen()) / ZZ(2)
a = c_km2_1_rho(k, rho)
return (ZZ((k1 - 1) * (k2 - 1)) / ZZ(60) + ZZ(c_km2_1_01(k)) / ZZ(4) +
ZZ(c_km2_1_11(k)) / ZZ(3) +
ZZ(a + F(a).galois_conjugate()) / ZZ(5))
def c_km2_1_sqrt2(k):
K = QuadraticField(2)
a = K.gen()
k1, k2 = k
k1 = k1 % 8
k2 = k2 % 8
k = (k1, k2)
if k in [(0, 0), (2, 2), (4, 4), (6, 6), (0, 6), (6, 0), (2, 4), (4, 2)]:
return 1
if k in [(3, 7), (7, 3)]:
return 2
if k in [(0, 7), (2, 3), (3, 0), (3, 6), (4, 3), (6, 7), (7, 2), (7, 4)]:
return a
if k1 in [1, 5] or k2 in [1, 5]:
return 0
if k in [(0, 3), (2, 7), (3, 2), (3, 4), (4, 7), (6, 3), (7, 0), (7, 6)]:
return -a
if k in [(3, 3), (7, 7)]:
return -2
else:
return -1
====================================================================
"""
import pytest
from sage.all import EllipticCurve_from_j, GF, QQ, QuadraticField
from sage.rings.polynomial.polynomial_ring import polygen
from sage.schemes.elliptic_curves.constructor import EllipticCurve
from sage_code.frobenius_polynomials import (
semi_stable_frobenius_polynomial,
isogeny_character_values_12,
)
K = QuadraticField(-127, "D")
D = K.gen(0)
j = 20 * (3 * (-26670989 - 15471309 * D) / 2**26)**3
# this is one of the curves from the Gonzaléz, Lario, and Quer article
E = EllipticCurve_from_j(j)
def test_semi_stable_frobenius_polynomial():
# test that we indeed have a 73 isogeny mod p
for p in 2, 3, 5, 7, 11, 19:
for pp, e in (p * K).factor():
f = semi_stable_frobenius_polynomial(E, pp)
assert not f.change_ring(GF(73)).is_irreducible()
def test_semi_stable_frobenius_polynomial_t():