curve #700

y² + xy + y = x³ − 44922712597935075769x + 117371553223926832084436978376

a-invariants: [1, 0, 1, -44922712597935075769, 117371553223926832084436978376]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 2083833210201936131575422855485481092297287900563444530
discriminant (Δ): -149264972846764685104747539138425010641254732317359531683900
Faltings height: 10.1580
naive height: 147.3934
primes of bad reduction: 2, 5, 13, 19, 29, 29091626555939356855722781732311616533537454984831
regulator: 3629494114507504.19842559720823021611976252903215276610638090748344596232
submitted by: David Renshaw
submitted at: 2026-09-10 16:08:28 UTC
last updated: 2026-09-10 16:08:28 UTC

Witness: 17 independent points log in to add more points →

(-4769299827, 472377120992838)
(-2474675755, -461935703049708)
(6217015513, -279970912418682)
(2889583548, 108122362618188)
(1140635809, 260028510502998)
(1134464609, 260515061627592)
(14615968567/4, 358753936466129/8)
(7118707421, -397907429439108)
(4836443978, -115048287890817)
(16180466187/4, 343523724458779/8)
(7931056153, -509870152723962)
(35422336528, 6555285091672183)
(13661614263, -1432994767958332)
(12980421758, 1311993727025298)
(41769878980/9, -2539945395041396/27)
(107659042489/9, -30686306064769528/27)
(73310937783, -19769538702562332)

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 16:08:28 UTC · history