curve #681

y² = x³ − x² − 226840098905x + 45234150343722525

a-invariants: [0, -1, 0, -226840098905, 45234150343722525]
rank (lower bound): ≥ 12
torsion subgroup: trivial
conductor (N): 190048676194583852124740520
discriminant (Δ): -136892061462958748685450596556000000
Faltings height: 5.4807
naive height: 90.2244
primes of bad reduction: 2, 3, 5, 7, 1257331, 5891203, 30544424821
regulator: 22908317.713693262913394908649707061460954505897941124148453559
submitted by: NDElkies
submitted at: 2026-09-08 19:53:10 UTC
last updated: 2026-09-08 19:53:10 UTC

Witness: 12 independent points log in to add more points →

(156635, 116387250)
(-394615, 270737250)
(214385, 80351250)
(182885, 99324750)
(330585, 79828350)
(604985, 359760450)
(264435, 61159350)
(239585, 68110350)
(345635, 90116250)
(233285, 70792050)
(278645, 60506670)
(302165, 65419410)

Commentary

Found in the output of the searches from earlier this year that found Curves 157, 158, 244 of ranks 12, 13, 14. This one is of interest for having many integral points (at least 650 pairs), thanks in part to large Tamagawa number 240 (with |Δ|/N = 7203000 = 2^5 3 5^5 7^4).

last edited by NDElkies at 2026-09-08 19:53:10 UTC · history