curve #330

y² = x³ − x² − 482886126837705164255195633736x + 128875500361225289360623432042214069669747136

a-invariants: [0, -1, 0, -482886126837705164255195633736, 128875500361225289360623432042214069669747136]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 76112461196808026563075717934065940819332588227906251909576775948240
naive height: 216.6623
Faltings height: 15.8578
discriminant (Δ): 31287757709910264613757109283395755219056531022112192673139216897563002547901499488793600
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 31, 37, 61, 1132627, 2771315443373, 58061085533848247634531022486933677533
regulator: 1521972441335808431.928851598137744911238709585530506718258709695380672789941
submitted by: 7fff-zip
submitted at: 2026-08-24 19:17:42 UTC
last updated: 2026-08-24 19:17:42 UTC

Witness: 19 independent points

(2290804687204732584/25, -3467126139839410434496670952/125)
(14577958168095021, -1758167039920555817686650)
(56016686080303825/9, -13178617360573382520013288/27)
(-657295472815786776/1849, 1271147590336516660872452040/79507)
(-306471756195812010/2401, 1614978607764236363908820814/117649)
(691151498047146860753/2042041, 5987426964310655713564791841996/2918076589)
(33337411143611276768508/87965641, 453783752815879059182557725526596/825029746939)
(383446365856044, -305150462427497032500)
(10994356009984432186257/25080064, 150382639071347861191995229169607/125600960512)
(494012030343406, 3299475131381881330830)
(1220418581612028, 36841145344897523404236)
(63381732662543522016246/49434961, 13978323337335148903175525789893950/347577210791)
(-515788542746545644644346/11161288609, 14494136715613833223569847065569193390/1179156657675023)
(16506961344829371228919884/7211915929, 64277291605080054680070367599590751900/612457536438467)
(588440415263721921086/1530169, 414860888817657392198431091210/1892819053)
(874515181660886139644/26569, 25855623015409995950200727146940/4330747)
(2336385763075826094709029/6228682084, 354966584859075458936360848632239685/491580047433448)
(1712362799748476, 65749266927887199760180)
(19624507648156935696/17161, 73609385995203769256077362600/2248091)

Commentary

New in this search: Mestre quartic family with roots [0,115,206,234,731,784], T=4287/7. 19 independent rational points certified exactly by Cremona-Brumer quadratic characters; quartic point box [100000000,50000].

last edited by 7fff-zip at 2026-08-24 19:17:43 UTC · history