curve #66

y² + xy = x³ − 271568164801421919805097494520335727505515190x + 1673523352045742769296938739782713519216640490554763586630258973092

a-invariants: [1, 0, 0, -271568164801421919805097494520335727505515190, 1673523352045742769296938739782713519216640490554763586630258973092]
rank (lower bound): ≥ 26
conductor (N): 440755853609146510472634852930653385420305551113633650634939171542380165741313856027971752379908017654932910 ★ record for rank ≥ 26
naive height: 318.5520 ★ record for rank ≥ 26
Faltings height: 24.4585 ★ record for rank ≥ 26
discriminant (Δ): 71895080707631236566227414257088729466603817414726942045972724673173251962604021806427843161273966232348946866110784725884544768000000 ★ record for rank ≥ 26
primes of bad reduction: 2, 3, 5, 7, 11, 13, 37, 61, 109, 3180882909509, 101223837495550397872963, 185290954722549979010823774145698082035847060397002762839016907
regulator: 64414287721104130201143827656.9267422425825753392963916482410113025800545012498701248313219
submitted by: David Renshaw
last updated: 2026-07-01 22:29:38

Witness: 26 independent points

(8171422342464806465484, -6968834778825385043474309573382)
(15506445321751452148524, -1091324356491763240297753031369862)
(11039908252424365794924, -144827641060783314905542709878662)
(-1124601528095626940916, -1406238603313519698489557881370502)
(15979026151674771866124, -1189134868428022585423319170045062)
(11497067226620401106124, -266453792604568993030594792645062)
(7964508116120761886124, -125826249902394562842770754145062)
(2057879788890780254844, -1059897903660876637259757866892102)
(11013564232941423844134, -136087530861067930848004367296452)
(10829312030749636504524, -51221935741228061266466491121862)
(5848751988213230056404, -534099270092912134067240836612782)
(11638411535858522295324, -298922500066105927047772771275462)
(6868334166980311706124, -363743509637318702264451883645062)
(7381103602441056808764, -266792452337957421936180167447622)
(-18419992393365020549076, -652667554475023102928617867030662)
(5493669297710229516324, -589422439885808591350951805382462)
(10874362643126096194524, -79405280481739553235221538851862)
(12784040435352607621644, -539538954823537014427509667959942)
(5007446790086282822724, -662736416997155792108337796316262)
(36311702775006993614724, -6300067037234134629601408866834462)
(7959119841967332599004, -127557175274976008347318287354822)
(-12920905145755852965276, -1739335347799715861327567809614462)
(-17861114371778492133876, -908840632427321863500516490045062)
(19507248497293445799354, -1949133775462458912736164210505272)
(-17316278094075829267956, -1087992116843026234002459666650502)
(33532502811165086659374, -5502006283885945731489116619434562)

Commentary

Provenance: Andrej Dujella rank-record table "Rank >= 26", Elkies (2006). The page lists this curve y^2 + xy = x^3 - 271568164801421919805097494520335727505515190*x + 1673523352045742769296938739782713519216640490554763586630258973092 and the 26 independent points submitted here.

last edited by David Renshaw at 2026-05-29 04:47:29 · history