def isogeny_primes(coeffs, **kwargs):
del kwargs["label"]
f = R(coeffs)
K = QQ.extension(f, "a")
kwargs = {**LMFDB_DEFAULTS, **kwargs}
primes, _ = get_isogeny_primes(K, **kwargs)
return sorted(EC_Q_ISOGENY_PRIMES) + sorted(primes.difference(EC_Q_ISOGENY_PRIMES))
def several_d(self):
"""Table 1 from the introduction"""
output_str = r"${d}$ & ${Delta_K}$ & {lmfdb_link_latex} & {possible_isogeny_primes} & ${time_s:.2f}$\\"
latex_output = []
for d in range(2, self.range + 1):
Delta_K, _, K, Klabel = get_smallest_good_number_field(d)
t_start = perf_counter()
candidates, _ = get_isogeny_primes(K, ice_filter=True, appendix_bound=200)
t_end = perf_counter()
time_taken = t_end - t_start
candidates = [c for c in candidates if c not in EC_Q_ISOGENY_PRIMES]
candidates.sort()
possible_isogeny_primes = ", ".join(map(str, candidates))
lmfdb_link = LMFDB_NF_URL_TRUNK.format(Klabel)
lmfdb_link_latex = r"\href{{{the_link}}}{{{my_text}}}".format(
the_link=lmfdb_link, my_text=Klabel
)
output_here = output_str.format(
d=d,
Delta_K=Delta_K,
lmfdb_link_latex=lmfdb_link_latex,
possible_isogeny_primes=possible_isogeny_primes,
time_s=time_taken,
)
print(output_here)
latex_output.append(output_here)
def test_interval(D):
K = QuadraticField(D)
if not K.discriminant() in CLASS_NUMBER_ONE_DISCS:
superset, _ = get_isogeny_primes(K, **TEST_SETTINGS)
# test that there are not too many primes left over
todo = set(superset).difference(EC_Q_ISOGENY_PRIMES)
assert len(todo) == 0 or max(todo) <= 109
assert len(todo) <= 2 or max(todo) <= 31
def test_rational_isogeny_primes():
x = polygen(QQ)
K = QQ.extension(x - 1, "D")
superset, _ = get_isogeny_primes(K, **TEST_SETTINGS)
improperly_ruled_out = EC_Q_ISOGENY_PRIMES.difference(superset)
assert improperly_ruled_out == set()
todo = set(superset).difference(EC_Q_ISOGENY_PRIMES)
# would be nice if we could automatically do QQ
# i.e. we could rule out 23 as well.
assert todo == set([23])
def test_galois(f, extra_isogenies, potenial_isogenies):
K = NumberField(f, "a")
superset, _ = get_isogeny_primes(K, **TEST_SETTINGS)
assert set(EC_Q_ISOGENY_PRIMES).difference(superset) == set()
assert (
extra_isogenies.difference(superset) == set()
), "Known isogenies are filtered out!"
upperbound = potenial_isogenies.union(EC_Q_ISOGENY_PRIMES).union(extra_isogenies)
unlisted_potential_isogenies = superset.difference(upperbound)
assert unlisted_potential_isogenies == set(), "We got worse at filtering"
assert set(upperbound.difference(superset)) == set(), "We got better at filtering"
def test_from_literature(D, extra_isogeny, appendix_bound, potenial_isogenies):
K = QuadraticField(D)
upperbound = potenial_isogenies.union(EC_Q_ISOGENY_PRIMES).union(
{extra_isogeny})
superset, _ = get_isogeny_primes(K,
appendix_bound=appendix_bound,
**TEST_SETTINGS)
assert set(EC_Q_ISOGENY_PRIMES).difference(superset) == set()
assert extra_isogeny in superset
assert set(
superset.difference(upperbound)) == set(), "We got worse at filtering"
assert set(
upperbound.difference(superset)) == set(), "We got better at filtering"
def main():
for D in square_free_D:
K = QuadraticField(D)
if not K.discriminant() in CLASS_NUMBER_ONE_DISCS:
superset, _ = get_isogeny_primes(
K,
bound=1000,
ice_filter=True,
appendix_bound=1000,
norm_bound=50,
auto_stop_strategy=True,
repeat_bound=4,
)
possible_new_isog_primes = superset - EC_Q_ISOGENY_PRIMES
possible_new_isog_primes_list = list(possible_new_isog_primes)
possible_new_isog_primes_list.sort()
if possible_new_isog_primes_list:
print(f"D = {D} possible isogenies = {possible_new_isog_primes_list}")
def test_cm_type_2(f, extra_isogenies, potenial_isogenies):
K = NumberField(f, "a")
superset, _ = get_isogeny_primes(K, **TEST_SETTINGS)
assert set(EC_Q_ISOGENY_PRIMES).difference(superset) == set()
# for p in extra_isogenies:
# assert satisfies_condition_CC(K, p), "Not of type 2"
pnip = sorted(superset.difference(set(EC_Q_ISOGENY_PRIMES)))
print("f = {} disc = {} pnip = {}".format(f, K.discriminant(), pnip))
assert (extra_isogenies.difference(superset) == set()
), "Known CM isogenies are filtered out!"
upperbound = potenial_isogenies.union(EC_Q_ISOGENY_PRIMES).union(
extra_isogenies)
upperbound = upperbound.union(bad_cubic_formal_immersion_primes)
unlisted_potential_isogenies = superset.difference(upperbound)
assert len(unlisted_potential_isogenies) <= 2, "We got worse at filtering"
if unlisted_potential_isogenies:
assert max(
unlisted_potential_isogenies) <= 109, "We got worse at filtering"
def main(is_galois, class_number):
the_url = URL_TRUNK.format(str(is_galois).lower(), str(class_number))
payload = requests.get(url=the_url).json()["data"]
for dat in payload[:10]:
poly = R(dat["coeffs"])
label = dat["label"]
print(label)
K = NumberField(poly, name="a")
superset, _ = get_isogeny_primes(
K,
bound=1000,
ice_filter=True,
appendix_bound=1000,
norm_bound=50,
auto_stop_strategy=True,
repeat_bound=4,
)
possible_new_isog_primes = superset - EC_Q_ISOGENY_PRIMES
possible_new_isog_primes_list = list(possible_new_isog_primes)
possible_new_isog_primes_list.sort()
msg = "poly = {} ; label = {} ; possible isogenies = {}"
print(msg.format(poly, label, possible_new_isog_primes_list))
def test_get_isogeny_primes(coeffs):
f = R(coeffs)
K = QQ.extension(f, "a")
_, _ = get_isogeny_primes(K, **TEST_SETTINGS)