curve #4085

y² = x³ − 1162025894298242675x + 482138511095590345548053250

a-invariants: [0, 0, 0, -1162025894298242675, 482138511095590345548053250]
c-invariants: [55777242926315648400, -416567673586590058553518008000]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/4ℤ
conductor (N): 40338783648400
discriminant (Δ): 34860776828947280000000 ☆ record for ℤ/4ℤ torsion, rank ≥ 5
Faltings height: 8.2297
naive height: 136.4037
primes of bad reduction: 2, 5, 29, 149, 1861, 12541
regulator: 404692.122756946067264148631661341987275148479135
submitted by: jesper-petersen
submitted at: 2026-10-09 05:51:49 UTC
last updated: 2026-10-09 05:51:49 UTC

(-658139677935/529, 17819207456082900/12167)
(274432976185/441, 28424118310300/9261)
(1379865229041635/2217121, 322652050/3301293169)
(6481047529077465/10413529, 56462939100/33604458083)
(87394986503870843730/140423322361, 5276022381065850/52620972011659891)

Small-discriminant Z/4Z curve found by searching Tate normal form with a Nagao-type sieve.

last edited by jesper-petersen at 2026-10-09 05:51:49 UTC · history