curve #5088

y² + xy + y = x³ − x² − 33519964642838343060020312285186708x + 2373899869327718611968847677131325135909338705479031

a-invariants: [1, -1, 1, -33519964642838343060020312285186708, 2373899869327718611968847677131325135909338705479031]
c-invariants: [1608958302856240466880974989688961969, -2051049487099148873500772030188382816461281187933554153]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/3ℤ
conductor (N): 1242428825390939691392468726268105015400663549890
discriminant (Δ): -24084668962373785968456972345263040447034191898777114920601832671018286783956200562069775323556400230400
Faltings height: 18.6784
naive height: 250.1159
primes of bad reduction: 2, 3, 5, 7, 13, 19, 29, 37, 41, 79, 173, 509, 617, 4091, 6367, 116201, 351529, 167683603
regulator: 7894877304163844.6441887863900360117169602506356949641717529757
submitted by: Maksym Voznyy
submitted at: 2026-10-21 02:27:19 UTC
last updated: 2026-10-21 02:27:19 UTC

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

(30132647849878073937/289, -17318034341235381005592397495/4913)
(6304621253763744516345595857/61735850089, -60959620332514625994563993201978197151465/15339321464063563)
(36475881073774046348042514192399/198203402011729, 137777658502087252195473442353732837347917414507/2790401243730272409817)
(-11640355085044234643083312392617/55060748205849, 1393095126450615067666190748021685508866067047/408566884486623893757)
(613017859614368133619238281286581/6839623285060225, 5353564380386647118685344474853871479557579884659/565650627515400003853375)
(909137062803733551368585398687/9565269571729, 201914578717057423542200489714377866165388725/29583245730243301433)
(259021344724276615799517005170293/1320126928269601, -2776444548630987437418112510606407682232814152845/47964899074756889683151)
(48765521673382641918343831/211818916, 255319774201401950732996425664594494839/3082812503464)
(13473464338366433084409/107584, 417842982834401889029721122061937/35287552)
(12832142631558573798807/151321, 699458924141814459597747857005289/58863869)
(-109814651154186466503/529, -239226327793364211603900579433/12167)
(86077370831720891842718927109/761216115529, 3602326594124378459939699125815879867829721/664143552828395333)

Commentary

t = 8643/3365 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 02:27:20 UTC · history