curve #4690

y² = x³ − 248621438602x

a-invariants: [0, 0, 0, -248621438602, 0]
c-invariants: [11933829052896, 0]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 3956007662881795565721856 ☆ record for quartic twists, rank ≥ 9
discriminant (Δ): 983548316266207850626632277313485312 ☆ record for quartic twists, rank ≥ 9
Faltings height: 5.5958 ☆ record for quartic twists, rank ≥ 9
naive height: 90.3312 ☆ record for quartic twists, rank ≥ 9
primes of bad reduction: 2, 17, 101, 229, 409, 773
regulator: 552244464.20081217659829337478907769392893950023801125089
submitted by: David Renshaw
submitted at: 2026-10-20 20:44:58 UTC
last updated: 2026-10-20 20:44:58 UTC

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

(-363609, 205736967)
(-729, 13462713)
(-17, 2055861)
(-17491098089/182329, 11798946038483073/77854483)
(-160815937/6889, 43512450521901/571787)
(-13399249/49, 74785354293/343)
(-3969704/9, 4169707028/27)
(43393673/16, 280978426971/64)
(85137849/64, 728297539629/512)

Commentary

Quartic twist y^2 = x^3 - 248621438602 x, rank exactly 9 (PARI ellrank 2-descent: lower = upper = 9). Found by a Mestre-Nagao sieve over all |a| <= 10^12 (a_p via quartic residue classes for p = 1 mod 4), top candidates certified with ellrank.

last edited by David Renshaw at 2026-10-20 20:44:58 UTC · history