curve #713

y² + xy + y = x³ + x² − 7137486693538189701x + 7305635188399970781241285899

a-invariants: [1, 1, 1, -7137486693538189701, 7305635188399970781241285899]
rank (lower bound): ≥ 15
torsion subgroup: trivial
conductor (N): 71322080623534602435688594522515972989490
discriminant (Δ): 214210224320635458920227685132913553198143841984900300800
Faltings height: 9.6533
naive height: 141.8493
primes of bad reduction: 2, 3, 5, 7, 11, 13, 37, 1620710158493, 39606055663157081210863
regulator: 230070913662.08765958342986104694261172996629511815758636901026422625
submitted by: David Renshaw
submitted at: 2026-09-10 16:08:49 UTC
last updated: 2026-09-10 16:08:49 UTC

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

(1668134619, 6419437660560)
(-3045194001, 28320315759660)
(4503539191, -257881795667572)
(216984919, -75941693932180)
(-2776507401, -75621642251140)
(2142922023, 43023359965692)
(1646800999, 4201884949340)
(1187434399, -22464364706860)
(1713922119, 10354569329820)
(1213205781, -20787000284490)
(41267259391/9, 7193358437535220/27)
(-2378407881, -104052906821940)
(1632215879, 2036143021980)
(2392302679, 62624861407340)
(1237314229, 19197903030230)

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 16:08:49 UTC · history