curve #65

y² + xy = x³ − 1222583105876029916237789137035775062690200x + 523967447200209449943328506898413682821590945806099349816040000

a-invariants: [1, 0, 0, -1222583105876029916237789137035775062690200, 523967447200209449943328506898413682821590945806099349816040000]
rank (lower bound): ≥ 25
conductor (N): 3421625153333343825198002959770328244842456471613462044687992344664286334910845403447307652180335890 ★ record for rank ≥ 25
naive height: 302.3562 ★ record for rank ≥ 25
Faltings height: 23.0460 ★ record for rank ≥ 25
discriminant (Δ): -1648077180048519919196851388268507997232089165506048976474499224670899793965899322864689351309245989194390271909250018304000000 ★ record for rank ≥ 25
primes of bad reduction: 2, 3, 5, 7, 11, 13, 157, 283, 72551, 895613, 39466373175449369221626360280383658386480681928740709773906484427935361743867171
regulator: 724084252210012759383791369.1859440608064513839867489159651105029300072050018474263574137
submitted by: David Renshaw
last updated: 2026-07-01 22:29:39

Witness: 25 independent points

(641919671491882025200, 1917622583611100541324074905000)
(594994690844383471600, 2678847964182260740553494589800)
(208196841608040479152, 16686940720261011217265226309352)
(-740361154329309511040, -31989105845069848078376802650600)
(804899774308127940400, 7834276239548635107854922077800)
(77301598762979192560, 20734554180656765176390619976040)
(702911299694560038040, -3449260992056554312601103761600)
(915485567326094606800, -13114547973847689345711009896600)
(-26293910519288470100, -23581682704464936496514064216200)
(709908819594746168740, -3717262868674905037217938483100)
(875280814115630686150, 11154863867288165850338319192550)
(1643784276034119529840, 54367762877007597564101217327400)
(2310209163245804946160, 100146314980252246302935050670440)
(1909368489723249677080, -71767455269710117770413293033280)
(673388810420663671600, -2458352439913512016092058261400)
(-1225141077322557861200, 13524179089651555542230916989800)
(641884454793213323650, 1917498326574315724474707751300)
(1779583257516825231520, 63119495622551336426454649548280)
(1336811684405040135250, 35757154248294043850600009373700)
(591774410894432534320, -2776985009258775118824482266520)
(4566516049692282828100, 300278030461559541447358362839800)
(-499965890434284752520, -31784318126596227394232274064800)
(1239030190644321769050, -30187812602932900034742078815100)
(3417029330608564213375/4, -81177702701038029224605519995775/8)
(9868611464482118831434/25, 1267809878380229048466900915382388/125)

Commentary

Provenance: Andrej Dujella rank-record table "Rank >= 25", Elkies (2006). The page lists this curve y^2 + xy = x^3 - 1222583105876029916237789137035775062690200*x + 523967447200209449943328506898413682821590945806099349816040000 and the 25 independent points submitted here.

last edited by David Renshaw at 2026-05-29 04:47:28 · history