curve #6402

y² + xy + y = x³ − x² − 10561875302339597837x + 12233022495299492663524408885

a-invariants: [1, -1, 1, -10561875302339597837, 12233022495299492663524408885]
c-invariants: [506970014512300696161, -10569331433657396595979736144145]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 3840861332482897807031772359312847844502330538
discriminant (Δ): 10758075343847102648809637286729399096654666350316108644352
Faltings height: 9.8776
naive height: 143.0250
primes of bad reduction: 2, 3, 13, 73, 139, 18083623, 89089283, 139790951, 7182657247932809
regulator: 349714321716417.769409700898702031133937063652232658783948281038957229480
submitted by: Andrii Prokhorenkov
submitted at: 2026-11-02 01:47:28 UTC
last updated: 2026-11-02 01:47:28 UTC

Witness: 17 independent points log in to add more points →

(3508501871, 135515279952632)
(69923449180831/3025, -579188081444817944536/166375)
(-64644454729/25, 18647603148446104/125)
(86598886559/25, 16398675873741488/125)
(1651883153119/625, 823916784214524728/15625)
(5902869167, 394416149577560)
(7572657263, 605394129656984)
(-118463852513/49, 52734187497621512/343)
(226967225911/25, -101906938895823656/125)
(18023639143, -2382628794543016)
(-4963905393217/1681, -9162057001201649288/68921)
(20034373781/16, 1994461586427383/64)
(155361380359/25, -53996277467266808/125)
(1207340747, 35229305373020)
(22728963727, 3393229984338680)
(-94783179809/49, 54693086949112520/343)
(1525691031775/289, 1581380831523109432/4913)

Commentary

Specialization at T = 139/35 of the elliptic fibration of X637 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 4200T^2 + 38T - 141, p = -407T + 3117, q = -6825T^2 - 7432T, D = 1055775T^3 - 16146795T^2 - 4120746T + 2212056, E = -1475250T^3 - 30473776T^2 - 36505848T.

last edited by Andrii Prokhorenkov at 2026-11-02 02:22:59 UTC · history