curve #743

y² + xy = x³ − 13483304063305650807189160692120045x + 599280263589397319336516618899204444269356349200337

a-invariants: [1, 0, 0, -13483304063305650807189160692120045, 599280263589397319336516618899204444269356349200337]
rank (lower bound): ≥ 21
torsion subgroup: trivial
conductor (N): 45338571294080722176878229370326088934964788929204379574261910884085409670
naive height: 247.3739
Faltings height: 18.4546
discriminant (Δ): 1733385586882379368494476030773071654575790685170142154042556620709636153875714478480507056455680000000
primes of bad reduction: 2, 3, 5, 7, 11, 17, 29, 43, 79, 1417216780984163857, 8269459002193140764341761594260750978809385681
regulator: 95359065071160900193.694518373815593526344579247838909774233441621416310292714223
submitted by: Valery Asiryan
submitted at: 2026-09-16 18:15:55 UTC
last updated: 2026-09-16 18:15:55 UTC

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

(-533719755788960769/4, -38224848201243041884427931/8)
(-38439035239881261204/289, -30933927645462784115782637691/4913)
(-120920185710088926, -21485489602278401653400097)
(-111413144306344986, -26805570302522205506907357)
(-65401811258844606, -34660687573115795950916577)
(-6624992471499486, -26235782376872039382925857)
(7369784481944514, -22367644022971252363333857)
(33967338182387514, -13434252964201562832852657)
(471830284751756626/9, -163118499258052265590990139/27)
(1367599802403035921181/21904, -2949760808712816650870433307269/3241792)
(71504436466585794, -870703794752315588766177)
(81554049290029602, -6487167028781214493385217)
(85711538805090114, -8560525219221954681417057)
(139305361788954354, -37740446679289345255319937)
(265234181194617474, -125227999737500033851985697)
(487601640204954114, -331594183488259751005353057)
(1734542422054581594, -2279434392197401309764235377)
(89830776010866114, -10628272742536628964345057)
(62629929195306114, -698773983246511953825057)
(130165708221295014, -32397971629514525792907657)
(1454763576490114554, -1749215349434241888157403337)

Commentary

Specialization T=-19/74 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 21 listed points are certified independent by exact quadratic-character certificates.

last edited by Valery Asiryan at 2026-09-16 18:15:55 UTC · history