curve #6085

y² + xy + y = x³ + x² − 2413547343280917749249419542960704279422894x + 1431615770838892557149676069244967765505961218584294918413701611

a-invariants: [1, 1, 1, -2413547343280917749249419542960704279422894, 1431615770838892557149676069244967765505961218584294918413701611]
c-invariants: [115850272477484051963972138062113805412298913, -1236916026004803169378189000871233279786880283892296663050030433905]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 781206292224258045638669835730061367474174860190703579746
discriminant (Δ): 14408762850756776453683650180256071785671783989261206338220897759663794435279933212025243954276628898613641631354873204979073024
Faltings height: 23.2210
naive height: 304.3826
primes of bad reduction: 2, 3, 7, 11, 17, 19, 29, 79, 109, 229, 269, 389, 449, 727, 773, 1327, 2179, 2731, 14897, 22817, 72469, 2546329
regulator: 480108167456158.332950486728226314081251059109030090022581534529627
submitted by: The Wren
submitted at: 2026-10-29 01:05:59 UTC
last updated: 2026-10-29 01:05:59 UTC

Witness: 14 independent points log in to add more points →

(14201051616169454692655, 1682581616478834228343277428053717)
(-1227761369698203306891, -50439577248985937030916337566665)
(-1432547749392475161293, 44150572412259747928989046205115)
(40350827152666016025198073/39601, -43346865913654250895503292312704440655/7880599)
(1376546732755023974781713, 1615050711646726951322471525576630295)
(9991153646687740465553, 987251707759339727754337418403735)
(1010247597795307038705, 4939056025235211519099884600407)
(2187919434689532519622531/2209, -370638346517058703393114225788857035/103823)
(1578857745036752095541, 39455436405665014533066149767479)
(9480269054941054985927129809/28547649, -3938507955592900429131543243180381067574327/152530088607)
(6464603827269514399345/9, 223602326075196084946250759262349/27)
(224546729260882360232737745/452929, -5758722490863736138841442862665502129161/304821217)
(71872310796946647104169/25, 16888388855583065082081322386480443/125)
(246085448561580066242714380443377/236535349801, 786166943423051513383149987919228970527299966123/115038730840366549)

Commentary

Original specialization/search by kaankolcu (2026), public Yukon elliptic-rank challenge submission 1da657b4-8209-460d-8ef5-accfd1c896f3. Source: https://raw.githubusercontent.com/Layr-Labs/elliptic-rank-challenge-public/f034f33d9fd851c472e99fe31ef5e489469b171a/submission/board/2x2-14-own.ec . Specialization T=86513/4358 of the rank-five X312 family posted by wgxli with curve#4269: y^2=x(x+A)(x+B), A=-q(29p+2q)(400p+49q)(221p+86q), B=-p(8p+53q)(25p+88q)(392p+221q). The fourteen supplied points were checked exactly on the displayed model and certified independent modulo torsion by an exact rank-14 F_5 reduction matrix at fourteen good primes.

last edited by The Wren at 2026-10-29 01:05:59 UTC · history