curve #336

y² + xy + y = x³ − x² − 15562580896796750296231806771530x + 23456109321321254485649290545253053437231113097

a-invariants: [1, -1, 1, -15562580896796750296231806771530, 23456109321321254485649290545253053437231113097]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 85455106033620726737244174703654827512709437318056845761537606762550
naive height: 227.0809
Faltings height: 16.7755
discriminant (Δ): 3544726544965643121274669432217026701297680718000813558390880368205080304016057400950784000000
primes of bad reduction: 2, 3, 5, 7, 17, 19, 31, 41, 61, 3449, 394811, 795549335102805422537844157414441015295531368511
regulator: 82887976042256042412.09408098706088009346633362934976932526983721497130669867
submitted by: 7fff-zip
submitted at: 2026-08-24 19:18:38 UTC
last updated: 2026-08-24 19:18:38 UTC

Witness: 19 independent points

(409691533901173609, 262219764785557112439941895)
(890231819944231696161/529, 26561553854378298458693996966465/12167)
(45567182897204134209, -307594289422575186875851084505)
(63175870291011535660593/160801, -15878337908118844106092342975151929/64481201)
(240896772069818642121/1849, -3737228019675740945874754477355/79507)
(-1708229991390495, 212263697073716737861831)
(-813229246067344725579/573049, 89622493920158626179994036768855/433798093)
(429243338708761/9, 4069212689107427777096845/27)
(102095509464973527609/175561, 8889067560037020782074352027885/73560059)
(431214922555557681/289, 293022391856009204424854855/4913)
(27152520564440015160369/7134241, 2651099281776349889901270083461945/19055557711)
(4873010421309969, 251664694024185370945015)
(13649932583307609, 1534370343091861450195495)
(13945021565629920361/729, 51044619867771013294954990085/19683)
(-4635737253241708335/43681, 1446545720515270792335518544167/9129329)
(69001266790065441961201/1420864, 18067275787972871281115362781545435/1693669888)
(2084323733451793, 8590688171130949818871)
(13798057092247569, 1560668075628647535937015)
(626511991436294731238324505/20204063881, 15560532783894119034948542124787342243123/2871825844109221)

Commentary

New in this search: Mestre quartic family with roots [0,115,206,234,731,784], T=14021/38. 19 independent rational points certified exactly by Cremona-Brumer quadratic characters; quartic point box [10000000,10000].

last edited by 7fff-zip at 2026-08-24 19:18:38 UTC · history