curve #6405

y² + xy = x³ − 1164519483789380670518459517436x + 485643117174787578417565198941268687348908560

a-invariants: [1, 0, 0, -1164519483789380670518459517436, 485643117174787578417565198941268687348908560]
c-invariants: [55896935221890272184886056836929, -419595653239016551598179164720664423198542251233]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/3ℤ
conductor (N): 924067205461080002728932377057018486310
discriminant (Δ): -817339230195339457068213486106403890793398374395209101272489394235993751707168205596262400
Faltings height: 16.1028
naive height: 219.3112
primes of bad reduction: 2, 3, 5, 7, 13, 19, 23, 31, 59, 107, 263, 421, 631, 56649331198003333129
regulator: 37840654675966.7259066073886520120271416834948251092992251553
submitted by: Maksym Voznyy
submitted at: 2026-11-02 01:57:09 UTC
last updated: 2026-11-02 01:57:09 UTC

Witness: 11 independent points log in to add more points →

(437898261936075873464/737881, -1196189679717190724188721138740/633839779)
(-15945207290586045511673032/18919727401, 76691792196382969544649275932065361684/2602389584280149)
(2267397934451472043837181048/12271282235521, 715158470690338560576960118421184631020604/42986780251037248319)
(9156962781441400606094456/13219710529, 5121645123523086754221847640087109700/1519962657492833)
(-35470402535108296/121, 37689257503107843188654476/1331)
(332555716928519588, -191776038825583231937222284)
(137417719585589160487551741092/85476737001769, -41581807330521802334066328284913483835970948/790263461636886047147)
(3077482318685240847008306/3236813449, -2840497288648462662439101474367658170/184152027553957)
(130540938229859864/961, -541149955153256688322720420/29791)
(4168569051331008179600617459832/135628920036289, 8505821809998324005788508324750593127450556092/1579532097050981077087)
(199850886778370216024837432/117007489, 2825263980473103216366267969777447308548/1265670008513)

Commentary

Voznyy (2026), t = 2209/947 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 2026-11-02 01:57:09 UTC · history