curve #694

y² + xy + y = x³ − 1479394361826583559070054x + 663363484630730621642552445021129556

a-invariants: [1, 0, 1, -1479394361826583559070054, 663363484630730621642552445021129556]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 508367396979061963745403246513051298909504003446659603681564671110
discriminant (Δ): 17118006258451433764836728272243184993043262651097679971563523203598186100
Faltings height: 12.8125
naive height: 178.5746
primes of bad reduction: 2, 3, 5, 31, 149, 6791, 136727, 231351918529, 17078406746663744724133429683950149793191
regulator: 14216995727906520014.78412702068255553203350056377588285617590229486588710478
submitted by: David Renshaw
submitted at: 2026-09-10 16:08:16 UTC
last updated: 2026-09-10 16:08:16 UTC

Witness: 19 independent points log in to add more points →

(1173856664483, 737745969993536733)
(-587414304517, 1153122282035595933)
(982582332314, 397980111844941792)
(-933982308757, 1109215252576345143)
(-20452357750, 832833746438049717)
(579677248913, 24055722547807803)
(3910172452247/4, 3112013199813520269/8)
(-1406999637673/4, 8542484437258272549/8)
(946727654993, -333655293485987637)
(10752842504291, -35043382285818747969)
(-656158210384, 1162573132424573931)
(-564625887664, -1148330573857756533)
(2356112319398, -3202679537590770312)
(2271322245947/4, -640234210182584421/8)
(10725068113625, 34906585575257213607)
(939935215666, 321310761730174781)
(-59388947459656/49, 282006438639515235276/343)
(1052418817136606/625, 26821936942848112408296/15625)
(133739744512817819/28561, -47393802295849637610821067/4826809)

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 16:08:17 UTC · history