curve #334

y² + xy = x³ − x² − 20088058719108362343982670142x + 1058707069906879133804800274913812097605016

a-invariants: [1, -1, 0, -20088058719108362343982670142, 1058707069906879133804800274913812097605016]
c-invariants: [964226818517201392511168166825, -914722908399539232586664110119267352073983125]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 4645654301378334775416336624703209894193008406665345269653301849346150
discriminant (Δ): 34580924922872053571701566010054140579738463663558732474684969795750563761232693187500
Faltings height: 15.1810
naive height: 207.1234
primes of bad reduction: 2, 3, 5, 13, 31, 53, 73, 511982641, 33062977231, 113353501142719, 3450630579549091047282711829
regulator: 62907649343052817485.60738489546805567900143908991350939103672986355899666704
submitted by: Andrii Prokhorenkov
submitted at: 2026-08-24 19:18:27 UTC
last updated: 2026-08-24 19:18:27 UTC

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

(10508859682607163, -1077194402470961252875491)
(1306578693049494, -46960996436311739108772)
(612662279994124, -14789097770206369260612)
(-5926627722346818141/4813636, 10992890250054193839950098105377/10561117384)
(67809425786269, 91340535227206609553)
(66171076936014, -138549750679457517282)
(58198476681330, -294507315659386444224)
(258844710229899531/1849, 79110808276044753053458071/79507)
(540593268632382099/2401, 331714850451843238145651712/117649)
(501455703792639, 10820369197059447709893)
(31967357275729324105/59049, 175062045796321834905628510898/14348907)
(9566512359620240160879/51566761, 713911854210074815335753692965188/370300910741)
(1110770293379349874, 1170675099301436955861820238)
(274850537550015, 4037384841201267216636)
(730138840811274, -19381193367213954467112)
(3731969799604017021/55225, -1285624425785137175703469074/12977875)
(56854032117244, -316849875652479180972)
(831128595891639653442999/16387584196, -865867845122614673376815589482897043/2097840203266744)
(186814718475324298539/4541161, 5317559327289518776055812906383/9677214091)

Commentary

New in this search: Mestre quartic family with roots [0,115,206,234,731,784], T=2194/13. 19 independent rational points certified exactly by Cremona-Brumer quadratic characters; quartic point box [10000000,10000].

last edited by Andrii Prokhorenkov at 2026-08-24 19:18:27 UTC · history