curve #710

y² + xy = x³ − x² − 729859381899210x + 7859607725017909321816

a-invariants: [1, -1, 0, -729859381899210, 7859607725017909321816]
rank (lower bound): ≥ 15
torsion subgroup: trivial
conductor (N): 742147693474684155569122124774755137519270
discriminant (Δ): -1803418895143482498032966763202654984171826100
Faltings height: 7.4505
naive height: 114.3552
primes of bad reduction: 2, 3, 5, 10499, 785416276126492634821434977696029397
regulator: 3294293008378.0187946469376956023484604342324432197990093408918971854
submitted by: David Renshaw
submitted at: 2026-09-10 16:08:44 UTC
last updated: 2026-09-10 16:08:44 UTC

Witness: 15 independent points log in to add more points →

(15528998, 16437207266)
(24830798, 71026341536)
(18765083, -27784427239)
(-9790492, -118598793064)
(-16511692, 124141811456)
(-4663852, -105648593284)
(10500260, 36786126116)
(3748982, 71943031298)
(9141302, -44181724600)
(57918758, -399879318814)
(42340973, -229942391434)
(-31094167, -22135469614)
(24228452, -66335405500)
(-1772833, 95645826785)
(-11957215, -121965782326)

Commentary

Found by a search of rational specializations of Mestre families, using finite-field scores to select fibers and rational-point searches to construct the witness. Point independence checked with the site verifier.

last edited by David Renshaw at 2026-09-10 16:08:44 UTC · history