curve #40

y² + xy + y = x³ − x² − 244537673336319601463803487168961769270757573821859853707x + 961710182053183034546222979258806817743270682028964434238957830989898438151121499931

a-invariants: [1, -1, 1, -244537673336319601463803487168961769270757573821859853707, 961710182053183034546222979258806817743270682028964434238957830989898438151121499931]
rank (lower bound): ≥ 20
conductor (N): 14143803817470093613466453371895612334117548667434143819728526037275098905399795800409355189270
naive height: 401.1305
Faltings height: 31.4627
discriminant (Δ): 536322869068731689907926014223523897218983690620200910078755027869276074296716149126908879546977218241763631376739573056546560007049072199214175253755603358448062903360000
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 23, 31, 79, 131, 137, 227, 229, 337, 443, 593, 751, 1093, 1297, 2113, 8231, 8629, 19937, 33073, 47419, 157559, 479081, 858259, 2391953, 11332579
regulator: 2345864571013562003762503.94206244554266939212953056905157720878181020686213493
submitted by: David Renshaw
last updated: 2026-07-01 22:30:40

Witness: 20 independent points

(-5976635286513806621064126789, -1486518161621811752215471892311622350764326)
(595416388787490259443766591, 903504029373113768509734316584679010931354)
(2434562872293108275107029075, 617088105910990842809058933920186352787402)
(3513074027344435171140978981, 382084415613590558839076708139976548338374)
(399682145249051758133327419, 929535737737305075763667416604801178363754)
(-10714754038296881855524018251, -1533542840817584957359022971329370697480444)
(-16034220456847626275437501599, -871976826237835562350053313866844829289386)
(1185828672355214392425799131, 820608248516937017244986013572105379548074)
(-11190697582885409770718510409, -1515528657208794070421971888182751388116946)
(2634316446310680332042122261, 579484021184216876019568884495793999507014)
(64222149978369055569434725591, 15815822465644868585416444171217848754979354)
(23945425437351916471937562579, 2972551260335618106366053972465493213480034)
(251112537077876411029986504415821/72361, 7660680492845270503732516891131042427064917041476/19465109)
(13094114400583295432756346651, 69052082853368511623611930199368498525114)
(2689776334541089917424552236511, 4411303346731260129651357011579276750542331014)
(-2627014038979941829331861469, -1259358775896763343051694775385987771517806)
(13434870681765045829611551147217379/1026169, 67909287018595777234819162150437056031048512222738/1039509197)
(-2128476924703349108908527744522501/243049, -186847352402051756015896357808242938299356659058862/119823157)
(-7364938748841807757773625709, -1537278043534293249189664536251504660398646)
(785686589410787916270883192839, 696287187804791686384919800041247659097028446)

Commentary

Found by Elkies - Klagsbrun (2020). A curve of rank exactly 20, via Dujella's elliptic-curve rank-records tables.

last edited by David Renshaw at 2026-05-27 22:32:12 · history