curve #3694

y² + xy + y = x³ − x² − 31334555664698466970327505x + 65603644346627699757540306732570171872

a-invariants: [1, -1, 1, -31334555664698466970327505, 65603644346627699757540306732570171872]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/4ℤ
conductor (N): 242574104114218087554953925 ☆ record for ℤ/4ℤ torsion, rank ≥ 9
discriminant (Δ): 109764042311428002918110138859392466498889591083535967987747292101080610390625
Faltings height: 13.5567
naive height: 187.7339
primes of bad reduction: 3, 5, 7, 31, 43, 47, 59, 73, 101, 137, 311, 353, 401, 937
regulator: 23495112.899865026232030790397532213802541551536589760340
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:44:41 UTC
last updated: 2026-09-30 01:44:41 UTC

Witness: 9 independent points log in to add more points →

(-4293314553312, 10999820947981814815)
(8486511603147/4, 23560597224186783305/8)
(4808975766624, 5111757697129571200)
(194304065100084/25, 2134374469033702018892/125)
(346502834659146/25, 5986793130328640969354/125)
(481437979901076/289, 20859843300714263004860/4913)
(739826936966181/169, 7713015403703935456825/2197)
(8203936891014204624/1437601, 14690046989164043479369046225/1723683599)
(47556559464052337469/17986081, 84865211962797690714859652525/76278969521)

Commentary

Specialization at t = 1553/5074 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 01:44:41 UTC · history