curve #10

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y² + xy + y = x³ − 120039822036992245303534619191166796374x + 504224992484910670010801799168082726759443756222911415116

a-invariants: [1, 0, 1, -120039822036992245303534619191166796374, 504224992484910670010801799168082726759443756222911415116]
rank (lower bound): ≥ 24
conductor (N): 325763846420202806803685538805031963849862314963649942610049440603050996133862755155678267754410 ★ record for rank ≥ 24
naive height: 274.6563 ★ record for rank ≥ 24
Faltings height: 20.7141 ★ record for rank ≥ 24
discriminant (Δ): 869228312577660772590804276974894693796770087691229179004716529760427780748163030521421801352204521776094281605900 ★ record for rank ≥ 24
primes of bad reduction: 2, 3, 5, 11, 13, 17, 29, 31, 41, 458619970494582607679296750333015081, 264240973182971699094661154229360236070105974082503
regulator: 104899016884108750894133329.51301636361085332121321081420962914892371133491485632339948
submitted by: David Renshaw
last updated: 2026-07-02 01:49:27

Witness: 24 independent points

(2005024558054813068, -16480371588343085108234888252)
(-4690836759490453344, -31049883525785801514744524804)
(4700156326649806635, -6622116250158424945781859743)
(6785546256295273860, -1456180928830978521107520473)
(6823803569166584943, -1685950735477175947351774817)
(7788809602110240789, -6462981622972389783453855713)
(27385442304350994620556, 4531892554281655472841805111276996)
(54284682060285253719/4, -296608788157989016192182090427/8)
(-94200235260395075139/25, -3756324603619419619213452459781/125)
(-3463661055331841724647/576, -439033541391867690041114047287793/13824)
(-6684065934033506970637/676, -473072253066190669804172657192457/17576)
(-956077386192640344198/2209, -2448326762443096987265907469107661/103823)
(-27067471797013364392578/2809, -4120976168445115434193886851218259/148877)
(-25538866857137199063309/3721, -7194962289937471269967128729589169/226981)
(-1026325011760259051894331/108241, -1000895294067489857736110963003267773/35611289)
(9351361230729481250627334/1366561, -2869749605748635777475372339306204832/1597509809)
(10100878635879432897339615/1423249, -5304965776276966451066900941489387801/1697936057)
(11499655868211022625340735/17522596, -1513435763341541188265230241426826478043/73349586856)
(110352253665081002517811734/21353641, -461706833308406671405570254542647784288/98675175061)
(414280096426033094143668538257/285204544, 266642138924791310663963499787603019833872421/4816534339072)
(36101712290699828042930087436/4098432361, -2995258855766764520463389153587111670142292/262377541318859)
(45442463408503524215460183165/5424617104, -3716041581470144108721590695554670156388869/399533898943808)
(983886013344700707678587482584/141566320009, -126615818387717930449161625960397605741940953/53264752602346277)
(1124614335716851053281176544216033/152487126016, -37714203831317877163580088877209977295481388540127/59545612760743936)

Commentary

Found by Martin - McMillen (2000). A historical rank ≥ 24 record, via Dujella's elliptic-curve rank-records tables.

last edited by David Renshaw at 2026-05-27 22:31:42 · history