curve #5664

y² + xy + y = x³ − x² − 150784916194219125998856683x + 711909032373952771446863141946724182331

a-invariants: [1, -1, 1, -150784916194219125998856683, 711909032373952771446863141946724182331]
c-invariants: [7237675977322518047945120769, -615089403971062624988191803310753940490753]
rank (lower bound): ≥ 21
torsion subgroup: trivial
conductor (N): 697797389882636893375354786566333572472674496043540960064894449988010
discriminant (Δ): 464761196852596238907527182158172513836443312600678754974729520236238176563200
Faltings height: 13.8128
naive height: 192.4473
primes of bad reduction: 2, 3, 5, 11, 73, 13204073, 1962705953277098415976674079, 372570212437022544795980655989
regulator: 121044529444239947725.84100718184407309359198785965999327534182145364066972982868
submitted by: apkorte
submitted at: 2026-10-24 01:37:59 UTC
last updated: 2026-10-24 01:37:59 UTC

Witness: 21 independent points log in to add more points →

(7665131721995, 2545936064620721162)
(10516749944195, -17009286836519332438)
(6711923043505, 1491242470849189592)
(4688446524553, -10393329426321637450)
(506933251167067/49, 5522714384669610302630/343)
(2811831119605, -17611326115728563548)
(18888139610395, -67841217489756084358)
(8931386517295, -8811609110652805738)
(857543097952951, -25109592063856002090994)
(-11151438261205, -31727669207523345238)
(9348263061511, -10921400518419443014)
(6900017694155, -44386833901560758)
(4610194924555, -10711948264155930358)
(-499442670595, -28055175228873330148)
(7488868847251, -1643163886883244094)
(6884782545995, -357710304147035158)
(6230029453855, -3784492495589215558)
(106476444682795, 1091698237782222694442)
(106176424803031/25, 1521322970837665425406/125)
(7304019101467, 482709357193238906)
(54723757679891, 395399849727113206082)

Commentary

Specialization at T = 7/17 of the rank-17 fibration of X1008 given in curve#751. Found by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-24 01:37:59 UTC · history