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test_frobenius_polynomials.py
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test_frobenius_polynomials.py
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"""test_frobenius_polynomials.py
Test for the code in frobenius_polynomials.py
====================================================================
This file is part of Isogeny Primes.
Copyright (C) 2022 Barinder S. Banwait and Maarten Derickx
Isogeny Primes is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <https://www.gnu.org/licenses/>.
The authors can be reached at: barinder.s.banwait@gmail.com and
maarten@mderickx.nl.
====================================================================
"""
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():
# an example where the frobenius polynomial depends on the purely ramified extension
# we make
x = polygen(QQ)
K = QQ.extension(x - 1, "one")
E = EllipticCurve(K, [49, 343])
assert E.discriminant() == -(2**4) * 31 * 7**6
assert E.j_invariant() == K(2**8 * 3**3) / 31
f1 = semi_stable_frobenius_polynomial(E, K * 7, 1)
f2 = semi_stable_frobenius_polynomial(E, K * 7, -1)(x=-x)
assert f1 == f2
@pytest.mark.parametrize(
"p, values",
[
(2, {1, 8}),
(3, {1}),
(7, {72}),
(11, {13**12 % 73, 57**12 % 73}),
],
)
def test_isogeny_character_values_12(p, values):
for pp, _ in (p * K).factor():
assert set(isogeny_character_values_12(E, 73, pp)) == values