curve #1680

y² = x³ + x² − 22953571400449597798612076573127126331566116785x + 1316112196573251143020836710988709170974662412826812302344253112205683

a-invariants

[0, 1, 0, -22953571400449597798612076573127126331566116785, 1316112196573251143020836710988709170974662412826812302344253112205683]

rank (lower bound)

≥ 27

torsion subgroup

trivial

conductor (N)

19227987025431238200907902998547750360067443762395378314231573179413790385819580948435378886999970282560339163766520

discriminant (Δ)

25692485156259326316214146811223863722063309088733532905224602482105266732792171899342965951042214877100781580662602903936209350062100000000

Faltings height

25.5433

naive height

331.8630

primes of bad reduction

2, 3, 5, 7, 11, 13, 17, 23, 29, 41, 43, 59, 6007, 47754823, 102454650789514599727, 5036174589815297392307, 916921362630975222773113542931195601712349410823

regulator

2646445789406381394608297578822.1825550226997542090412855677649877485153965650543438001525462

submitted by

Ritik Jain

submitted at

2026-09-25 04:08:41 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-25 05:07:59 UTC

Witness: 27 independent points log in to add more points →

(9054822081305276915436230381107991567524171532735995382111/90000760164928177504723290292884121, 135412887202370736788100409575440486381174164292645142798729670810563903032428903205010/27000342074939992257857752891510983362379120008756419)
(136081474401568992413859547/23409, 123180981173480547169629975154939706733646/3581577)
(9754532036064732485485286, 30461941347670840054013232746903657325)
(2752390880925821550121405019/49729, 162693346309167675166143414528401264771550/11089567)
(12768983768781966590725739/49, 39154373060059163723236787491137731250/343)
(534917215393727711845598819/5329, -1884441894625657440704195478229030224750/389017)
(246901878793628164611930011/10201, -28677368586558426421211953666709873307750/1030301)
(172403139389005098690784799/2209, 27636393056251116747129732358764200550/103823)
(852963768329041473041747399/289, -24879066287370809442993168492725610242970/4913)
(1413434795124125273425174/9, -1074843890749416054233874936725669375/27)
(41221427470334112330700271/361, -93028852504912037689799131379637629250/6859)
(416784308649580426721750351/361, -8438864223976752700246884582685083286590/6859)
(6511575072047699357061782/49, 8489476218789358689008026039425898221/343)
(-581880276380156096452672714/3721, -7464522977815756847320654610388315107535/226981)
(9651068905662421364498549/4, 29924414493421122723031182177717181905/8)
(6756887022467762635642769/49, 9536568664856742554801909589907756830/343)
(307820818356218924963113220529099468089999/622589097296181409, 163542096745579117339312257577609310815355753414976606980895450/491249664354002753817342577)
(541310848523361833300789193459389076881/17932295456533441, 61256960286398923012587023673321756201668953985260680157890/2401340959605782471673761)
(-2854782364449129477987292434552421418217364384974355501/35966624439073659080846482185409, 11078548368323962757778919261897929856385106849189940183069572636666420005531689950/215699689582928385832069818501384203149834739423)
(336218125790860075561283601001032641872271/5299074264678056041, 130908092691304149970721815787850269899435967342546564568553410/12198319634675180643440820661)
(-25084228901682660670885610195371756423/217756151572969, 158454723095300831579658148083911678953027262726068523094/3213332369324066145547)
(-16336899283429833883233872354175595621/752359202313889, 876579622518523351431475458624160160079872994847902124350/20636586204254447800463)
(91021768885475155368906779/2209, 2178479924489097079241522209482751322430/103823)
(609435058263461075832033819106954357937855161/408606725651102468493201, 9351515852780002302288808722306238594400315661929756666752916656337050/261191034941293528790035014627920199)
(153100157625469947389184785978019773693138313239/2024735076560996233892929, 10314389685707114620690272528592693973768100873353225572692150210605370/2881060047329908635370565675861719967)
(44838719197654012422707820467619231343588316441/464330842123152067528336, 75157221997346442566141268989848385157648131430765623121591238377285/316403513502795691953045360224267584)
(36361995634978210920278591065767481/7003771703296, 641298160468388104498071215811363058697639245635679725/18535229676991545344)

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