curve #11

y² + xy + y = x³ − x² − 20067762415575526585033208209338542750930230312178956502x + 34481611795030556467032985690390720374855944359319180361266008296291939448732243429

a-invariants: [1, -1, 1, -20067762415575526585033208209338542750930230312178956502, 34481611795030556467032985690390720374855944359319180361266008296291939448732243429]
c-invariants: [963252595947625276081593994048250052044651054984589912081, -29792112590906400787516499632162945722111222184084598858916705933795305936274003718521]
rank (lower bound): ≥ 28
torsion subgroup: trivial
conductor (N): 3455601108357547341532253864901605231198511505793733138900595189472144724781456635380154149870961231592352897621963802238155192936274322687070
discriminant (Δ): 3581775273581124569703746471783026864293570422220790957452423023734755536293093609524369063657264581131816964723217401534371110750621252297523717205467915359744000000
Faltings height: 30.6235
naive height: 393.6297
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 48463, 20650099, 315574902691581877528345013999136728634663121, 376018840263193489397987439236873583997122096511452343225772113000611087671413
regulator: 38572982340116091955788421590086427.57129683705100007574236938040229642098246623022300921177646
submitted by: David Renshaw
submitted at: 2026-05-27 22:11:54 UTC
last updated: 2026-07-02 01:49:25 UTC

Witness: 28 independent points log in to add more points →

(-2124150091254381073292137463, 259854492051899599030515511070780628911531)
(2334509866034701756884754537, 18872004195494469180868316552803627931531)
(-1671736054062369063879038663, 251709377261144287808506947241319126049131)
(2139130260139156666492982137, 36639509171439729202421459692941297527531)
(1534706764467120723885477337, 85429585346017694289021032862781072799531)
(-2731079487875677033341575063, 262521815484332191641284072623902143387531)
(2775726266844571649705458537, 12845755474014060248869487699082640369931)
(1494385729327188957541833817, 88486605527733405986116494514049233411451)
(1868438228620887358509065257, 59237403214437708712725140393059358589131)
(2008945108825743774866542537, 47690677880125552882151750781541424711531)
(2348360540918025169651632937, 17492930006200557857340332476448804363531)
(-1472084007090481174470008663, 246643450653503714199947441549759798469131)
(2924128607708061213363288937, 28350264431488878501488356474767375899531)
(5374993891066061893293934537, 286188908427263386451175031916479893731531)
(1709690768233354523334008557, 71898834974686089466159700529215980921631)
(2450954011353593144072595187, 4445228173532634357049262550610714736531)
(2969254709273559167464674937, 32766893075366270801333682543160469687531)
(2711914934941692601332882937, 2068436612778381698650413981506590613531)
(20078586077996854528778328937, 2779608541137806604656051725624624030091531)
(2158082450240734774317810697, 34994373401964026809969662241800901254731)
(2004645458247059022403224937, 48049329780704645522439866999888475467531)
(2975749450947996264947091337, 33398989826075322320208934410104857869131)
(-2102490467686285150147347863, 259576391459875789571677393171687203227531)
(311583179915063034902194537, 168104385229980603540109472915660153473931)
(2773931008341865231443771817, 12632162834649921002414116273769275813451)
(2156581188143768409363461387, 35125092964022908897004150516375178087331)
(3866330499872412508815659137, 121197755655944226293036926715025847322531)
(2230868289773576023778678737, 28558760030597485663387020600768640028531)

Commentary

Found by Elkies (2006). A historical rank ≥ 28 record, via Dujella's elliptic-curve rank-records tables.

last edited by David Renshaw at 2026-05-27 22:31:43 UTC · history