curve #67

y² + xy = x³ − 55671146865244401916117773020296610079754015500970x + 161981895322788558220906653027519611838007321625214218991719656790551905956

a-invariants: [1, 0, 0, -55671146865244401916117773020296610079754015500970, 161981895322788558220906653027519611838007321625214218991719656790551905956]
rank (lower bound): ≥ 27
conductor (N): 44473459337320533154136751324343533168070733706606064610068741773097621300831711931623980390498751106487256208641524555812214 ★ record for rank ≥ 27
naive height: 355.2704 ★ record for rank ≥ 27
Faltings height: 27.4823 ★ record for rank ≥ 27
discriminant (Δ): -292295992467036730780372591065803086322380999542983847983425194363595329691796423905092278238635719085287829796511182852120658943291432879857572200448 ★ record for rank ≥ 27
primes of bad reduction: 2, 3, 7, 13, 19, 23, 29, 43, 97, 1333872817, 622918016143240238107, 4018426744559269274579405699, 461516545359925528236128783904359556661401233959492820033
regulator: 70127565624872862829955299377887.957802713297922693486537158452665203606944461202704437006234
submitted by: David Renshaw
last updated: 2026-07-02 01:49:26

Witness: 27 independent points

(3767967516008165080365044, 2389736302094908158004099904947501190)
(6870254134405565034404108, 10187524517965617942800545361683736678)
(3887185284020449623939380, 2077020998301366905747533719381033926)
(4704247833799635063001076, 2048360739972031724784820678863578246)
(4126561570009022393013236, 1587663907962563318996362180056025478)
(4589477829219012602846900, 1774818405716699582839388275297252934)
(1744288391661626065189796, 8377495495389391047035879698795823126)
(375965292932773063399988, 11878746522289663117790823052090948358)
(-4430058725939313297140384, 17935065674772418581237173320631279206)
(46029381695079838296565796, 308418721198583803941973238472690797126)
(5015368619774521542769364, 2987769291318668561101046595511063430)
(55141979583089031946559900, 405905110011451276640435700460385551166)
(2703830808220294466353748, 5587793124284970779400186615144247334)
(3412724629872318338319668, 3426156011058008602456511184805561094)
(272723117214107051072886140, 4502171870151657762741942725666306991014)
(4732850534022088572670964, 2124602225002897987491873188898646406)
(19225480790209113087907256, 78725996092378368618479740248297817478)
(5213267756598937117846508, 3666105463387143768198032469471386414)
(-4503215618194252049902522, 17926532987110694852440283715314002874)
(10358928712485769814651816, 26398450763063898266637186797421380678)
(6560446866541184312028656, 8894515448962144734398280820434671978)
(4667249764662401626929236, 1954092716090144351072720616414325286)
(3131745787384349113625300, 4283649283716227803355987840842617734)
(243907731994687263474127628, 3807478665185691587984635270031859346574)
(110171466072672245507182388, 1153803508275547153736593941741941166854)
(2631452133741740392491152, 5805818938673165314161211507146370634)
(2398961346477899287733092916, 117498623151243646059583140149253976390406)

Commentary

Provenance: Andrej Dujella rank-record table "Rank >= 27", Elkies (2016). The page lists this curve y^2 + xy = x^3 - 55671146865244401916117773020296610079754015500970*x + 161981895322788558220906653027519611838007321625214218991719656790551905956 and the 27 independent points submitted here.

last edited by David Renshaw at 2026-05-29 04:47:31 · history