curve #749

y² + xy + y = x³ + x² − 42587034068961209637392019006781930558140x + 3341367072693013750934050754755346768524555742818154888613405

a-invariants: [1, 1, 1, -42587034068961209637392019006781930558140, 3341367072693013750934050754755346768524555742818154888613405]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 177333897875358942044372289057029604297059583650472500267542082220662532175676168235349290979480373230777830
naive height: 292.2707
Faltings height: 22.2299
discriminant (Δ): 120080217677316199668311097724447725701536431329135427246597561632874347406635426151185898153213667707949647047402956800000
primes of bad reduction: 2, 3, 5, 7, 43, 61, 17771167, 18115829590295440219231979319040439007168626893005566216203541377049492920519952312934723330203
regulator: 28514027801648985702331644406.56323143521063600484270591458480527697299074544975808256407
submitted by: Valery Asiryan
submitted at: 2026-09-16 19:04:54 UTC
last updated: 2026-09-16 19:04:54 UTC

Witness: 25 independent points log in to add more points →

(-189441883131138719987, -2147184594346231815138766005607)
(483256807419824265009963/5476, -210700312425978448380032984790859143/405224)
(90457422360057469803, -478775205565810271312789079147)
(107514582623956255813, -73772518679567048254518045607)
(107999268203006405913, -41048160083546936554802198307)
(108076781257086245053, -33138309177093372321771140647)
(31272624462509093755667/289, -49234347260348287235143018092411/4913)
(129748145526059036053, -5837717919601640642915876647)
(131676149652522536343, -129427949697198757600023867837)
(133415672227739212453, -185347740016739450379579266047)
(133500519231930433613, -187813867208059521583810196007)
(148368962485853212525, -537466993314064889597647641031)
(14475121336876610652865/81, -874159249294131080845528914801155/729)
(268469876878051854763, -3355336973460186908413606208107)
(303842737101605275513, -4295636367843411609337252412407)
(391290734749490669613, -6825494037490367655566643152007)
(397921014417244232329, -7028680945208438755326313214455)
(539592801503361469693, -11724749948670542250503106516007)
(366334865647105750909173838986613/431424335241, -6821078623207102194035299763254957654490024081443/283372014692010789)
(1852292336521665314943, -79244233050977785884999701486437)
(3628887097030723576813, -218258939066622151565352746625607)
(11261607309081721139123, -1194890608709439285787291068736107)
(16594151655441227508853, -2137465704832909283553558988538407)
(62948605226637700924013, -15793443705369688111000685403665607)
(88620677539930714190413, -26381617700406823963926200911437607)

Commentary

Specialization T=-332/267 of the family identified by wgxli in the commentary to curve #302. The points were obtained by specializing reconstructed generic sections and running minimized quartic searches. The 25 listed points are certified independent by exact quadratic-character certificates.

last edited by Valery Asiryan at 2026-09-16 19:04:55 UTC · history