curve #6191

y² + xy = x³ − 119801991309757280224082762725876x + 504903830340919253707299361454294229524627150480

a-invariants: [1, 0, 0, -119801991309757280224082762725876, 504903830340919253707299361454294229524627150480]
c-invariants: [5750495582868349450755972610842049, -436236909414554243828850022599034390443236774277793]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/3ℤ
conductor (N): 8286274652757291070116518242424271719730
discriminant (Δ): -83394833937982002001584379264645216187821964016393026831305398639950375461003274662279800422400
Faltings height: 17.1729
naive height: 233.2045
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 31, 73, 97, 211, 227, 271, 117721, 28016012371944271
regulator: 2499288656.92920652483135783332031743746919899566703495785646
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-30 15:39:57 UTC
last updated: 2026-10-30 15:39:57 UTC

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

(-10818419459289496, 731302739156452064070860)
(-5617392527237272, 1000310587001780845141196)
(4107922084895848, 286509801625647625925836)
(4207309335114104, 274472724255912790908236)
(6305969564742248, 13949518252940993349836)
(6418472761602848, 19455161837760161685836)
(8192443057331024, 270698825446377471099620)
(8332723222370480, 291896555982341022396620)
(17186071910291048, 1876719134429111775429836)
(21943796647536164, 2905614266881716742401440)
(31451885859761768, 5277288602520047698291916)

Commentary

Specialization at T = -961/194 of the elliptic fibration of X648 given by: y^2 + (Lx+B)y = x^3, where L = 3885067921T^2 - 14778309212T + 14171920816, B = 4936839685420991948803368000T^5 - 37762139727356747035718880000T^4 + 97681596151298882036363082000T^3 - 104946797327065126983361746000T^2 + 40090501217702000033914176000T.

last edited by Andrii Prokhorenkov at 2026-10-30 15:39:57 UTC · history