curve #3617

y² + xy + y = x³ − 7014056479127717x + 221483537524540059698900

a-invariants: [1, 0, 1, -7014056479127717, 221483537524540059698900]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ
conductor (N): 2485766715136584389804778
discriminant (Δ): 892767380867418374976295265370863566742778512844
Faltings height: 7.9865
naive height: 121.0737
primes of bad reduction: 2, 3, 11, 23, 71, 73, 227, 277, 317, 383, 467, 3853
regulator: 977218788.8778173703600762134710209857098997980636729842946
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:20:22 UTC
last updated: 2026-09-29 19:20:22 UTC

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

(-75590370, 565513296979)
(-1901412, 484576305403)
(-802688489661/9409, 407149773618562492/912673)
(-13342028085/169, 1169204415379060/2197)
(-7882795431/289, 3078078791533862/4913)
(-1610480772/49, 221369589080653/343)
(386510805/16, 16458857793535/64)
(61657592553/1681, 8024607878461088/68921)
(514381463955/47089, 3906605042607795772/10218313)
(127140579684279/2992900, 69846704323751412473/5177717000)

Commentary

Specialization at T = 10/23 of the rank-9 fibration of X400, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #802 (same T), 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 2026-09-29 19:20:22 UTC · history