curve #6414

y² + xy + y = x³ − 1519659876847126582785159x + 711752212806788927237642176798674046

a-invariants: [1, 0, 1, -1519659876847126582785159, 711752212806788927237642176798674046]
c-invariants: [72943674088662075973687609, -614953911865175048644455833868014907373]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/3ℤ
conductor (N): 414850321299557480517309911988570
discriminant (Δ): 5757464444255877554859411996575730901726207849377939773138921716885326300
Faltings height: 12.7643
naive height: 178.6552
primes of bad reduction: 2, 3, 5, 7, 13, 23, 37, 101, 127, 139, 181, 307, 389, 457, 6607, 219217
regulator: 111929954.5766176467420428023844376990327264283089471711010
submitted by: The Wren
submitted at: 2026-11-02 13:32:39 UTC
last updated: 2026-11-02 13:32:39 UTC

Witness: 10 independent points log in to add more points →

(537979222703, 223404307043641158)
(-375367431673/4, -7390949853275973651/8)
(631543126253, -62533250986696692)
(1826733249965, 2007849798734572812)
(15796996413809/25, 7707281147894964138/125)
(397029522383, -413506076024934822)
(3152443688459/4, 480143246322695523/8)
(-1402730294017, 288686864121826878)
(570535498838, 174492545916395253)
(105239076902453, 1079532842228677924158)

Commentary

Specialization at t=466/77 of the Z/3 K3 family y^2+(7t^2+309)xy-91(t^2-1)(t^2-25)(t^2-121)y=x^3, as posted in the Layr Labs Elliptic Rank Challenge repository (submission/board/3-r12-z3k3.ec, commit c8f82607a219df846ab80dd9e14ee09a2904d709). Sections from the same family, ellrank with 10 independent points, torsion Z/3Z (elltors), PARI/GP 2.19. Rank lower bound 10 is computed from these points; no upper bound claimed.

last edited by The Wren at 2026-11-02 13:32:39 UTC · history