curve #5091

y² + xy = x³ − 4552856386677926109892316347271480x + 118732085302285619788555986337690851171257777294400

a-invariants: [1, 0, 0, -4552856386677926109892316347271480, 118732085302285619788555986337690851171257777294400]
c-invariants: [218537106560540453274831184669031041, -102584521701174775825118032036575575324213496585908161]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/3ℤ
conductor (N): 12586515643015858148783450187219322636466538690
discriminant (Δ): -50108167234921189007404640083842306287894406653040397911321447775855739386831760747912581632000000000
Faltings height: 18.1716
naive height: 244.1251
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 53, 61, 97, 433, 509, 821, 953, 2109979, 44950014478843
regulator: 549841813752594.01099186510890041379890114296197122511952195756
submitted by: Maksym Voznyy
submitted at: 2026-10-21 02:32:34 UTC
last updated: 2026-10-21 02:32:34 UTC

Witness: 12 independent points log in to add more points →

(178341872994466086207670/5480281, -28783992553130306250163635358126130/12829337821)
(9110132810087690877335785485040/230805766597849, 2529821588904363922202479808265818588366627240/3506468832244135177757)
(-912538875437308105865626025095040/30405772592445121, 80117079501092792360213625239547931535089322760520/5301930915839635378468831)
(-261154018133034767617178910/3362028289, -374787388971973684343937977780063572410/194940486281087)
(-87919780026226668560/1849, -1199765482359540565040191966760/79507)
(7360850493699308041217440/161366209, 5007360911899749942588329559787713640/2049834952927)
(26466011441881141886910987835/567150572836, 1199853996514005972009911232818528409360505/427117693499354584)
(876167296340788450785570698245360/342897843285001, 25925708164868923052303120994706298167058286617320/6349611155927849302501)
(46747289048240350555208560/1248986281, 38453368401588045199299002392411769320/44140424156821)
(35534127212488480, -1348455731997431521128680)
(87017974816973229421855160944/2607885861025, 8387047538316015593500681780462465624347832/4211461837539967375)
(1051564996885618702059825014512/27262506265201, 101328079071660727729017959880968873621090296/142347114350313506551)

Commentary

t = 8945/2479 for y^2 + (7t^2+309) x y - 91 (t^2-1)(t^2-25)(t^2-121) y = x^3 from https://github.com/Layr-Labs/elliptic-rank-challenge-public/blob/main/submission/board/3-r12-z3k3.ec

last edited by Maksym Voznyy at 2026-10-21 02:32:34 UTC · history