curve #705

y² + xy = x³ + x² − 306367847659780934x + 52804619400469591847863104

a-invariants: [1, 1, 0, -306367847659780934, 52804619400469591847863104]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 185314350192848860902390821502956494336501428486
discriminant (Δ): 635830955060582569549027733313865009979004032269743684
Faltings height: 9.0400
naive height: 132.4043
primes of bad reduction: 2, 3, 23, 47, 1627, 146944121, 18100330733, 6602460748812427948591
regulator: 321501727065476.6552983339897969047194131403802696085271721787099551569
submitted by: David Renshaw
submitted at: 2026-09-10 16:08:37 UTC
last updated: 2026-09-10 16:08:37 UTC

Witness: 16 independent points log in to add more points →

(152201045, 3114546530147)
(1309264000, -43543647265848)
(1490353120, 53911310869472)
(1151367735, 35018997576003)
(193249720, -903488706292)
(144912736, -3384025844784)
(-430410989, 10243921291523)
(-5218144466/9, 160948666443532/27)
(-624901933, 478600371242)
(-6148466479/25, -1330377631562684/125)
(140250625, -3549027783277)
(-563930035/16, -510231671590237/64)
(4989302456, -350321838012772)
(747485279/4, -11535649655597/8)
(619150921, -10023625756284)
(4608639278, -310689152696386)

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 16:08:37 UTC · history