curve #3701

y² = x³ + x² − 1636853509520156312340125016033x + 854174720145841634402113003951554672311016063

a-invariants: [0, 1, 0, -1636853509520156312340125016033, 854174720145841634402113003951554672311016063]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/4ℤ
conductor (N): 7991717873699469142558439024164800
discriminant (Δ): -34514769803219878082821895968549422402914288669439600325923932047485735225123096668160000000
Faltings height: 16.3146
naive height: 220.4406
primes of bad reduction: 2, 3, 5, 7, 13, 37, 41, 53, 67, 107, 151, 163, 283, 3049, 19231, 77719
regulator: 17450379108.89054335994595193672390823791344088125802214245
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:45:59 UTC
last updated: 2026-09-30 01:45:59 UTC

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

(-992059195235427, 38751310189332827723700)
(53934870329523, 27677574555838474784400)
(26733781602981363, 4366187929960852682060400)
(10340308907073844713, 33250659707602573267581786600)
(-8787936372180468918/10609, 44274727264881876234043455975/1092727)
(14363366603964907/9, 1296756532157386938367600/27)
(9265390560997861827/7921, 16381844596182561596791187952/704969)
(14965997361820245907/1089, 57655456534678580468063009200/35937)
(19526720333481905883/19321, 40921698312333192971687019600/2685619)
(67613735376705890427/76729, 208293227864501801473648669200/21253933)

Commentary

Specialization at t = -2333/13936 of the Elkies-Klagsbrun Z/4 fibration E1: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = (8t-1)(32t+7), c = 8(t+1)(15t-8)(31t-7) (arXiv:2003.00077, eq. (5)), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-30 01:45:59 UTC · history