curve #402

y² + xy + y = x³ − x² − 12726070774451041281283293178772853133737609104772x + 17471855868336653804432728505912897769345397163476843729857408410639782695

a-invariants: [1, -1, 1, -12726070774451041281283293178772853133737609104772, 17471855868336653804432728505912897769345397163476843729857408410639782695]
rank (lower bound): ≥ 27
torsion subgroup: trivial
conductor (N): 237304270141300244936986435356903981816908015707178195445203338482306613950770931515606275822943007314335894 ★ record for rank ≥ 27
naive height: 350.8168
Faltings height: 26.9352
discriminant (Δ): 30718966157176916081764259804315592661235393543367655205731948923425663880417848596135494067342402515107622869752129282315055918823021902568620032
primes of bad reduction: 2, 7, 11, 13, 17, 23, 29, 31, 53, 67, 71, 83, 977740811, 23115571996982431216703381479827149860878061197218796523046010955046392792331767
regulator: 68327165255854520255801085457.495988916697010049962193254450312464719990291729796961760307191
submitted by: NDElkies
submitted at: 2026-08-28 16:53:47 UTC
last updated: 2026-08-28 16:53:47 UTC

Witness: 27 independent points

(1663112828519697244221957, 952387380417177680623422786669067197)
(2037930976023429894123141, 29340645544565289167196016943363261)
(1911208881800274197733445, 361643629052372259587725020124167549)
(2700495003049109528461701, 1673019137669994894510628551011395517)
(2232699933767023706197637, 433882221787588407421949864089404605)
(2114681435067716791945093, 129872018812746634398653182882940349)
(2154839281596420743368469, 234208377966574050508275922520222605)
(2037675218215549559398085, 30486948171391799093924069719696189)
(1567841483619081696962053, 1171899116109408154361721106817237949)
(1936737439382873351869069, 299010265565203861986244784697498445)
(2111515409980757536816485, 121437761927893771392924599448542589)
(-597287008991702417781969, 4985969255732457508122227705859928941)
(4104545027853244706504197, 5864092877843921415765474260949382077)
(1439687286898575798471813, 1460935542364728644809377742665755837)
(1927058769038959019380341, 322814616020344750309009962081331997)
(2135665420136391440248453, 184767953288226255564340485998353597)
(1738729312461290804027525, 775341605983083041066984435742295229)
(2039657535071554300250773, 20465408915088181089412630366993677)
(-3876666328912645897880827, 2923344637667261352831118736821679549)
(2085688033860396289096197, 46717623908600297259946937918257725)
(2017856627507653448498407, 93102055269992666832253889161049245)
(2012453481355446686317185, 107725007169415086605950410848234789)
(1638950951666961623631365, 1008431908308683071239002529721746365)
(2260433709356990115572741, 505212251696490591932006134783524797)
(2756243298611877034532997, 1826075089945216879758760488860031677)
(1886391005578580025321221, 422111549210750158601747577411709757)
(1964486185273722199878213, 230266058048637162094590586846957437)

Commentary

Noam D. Elkies and Zev Klagsbrun, January 2026

last edited by NDElkies at 2026-08-28 20:42:22 UTC · history