curve #5593

y² + y = x³ − 23909339438591055892

a-invariants: [0, 0, 1, 0, -23909339438591055892]
c-invariants: [0, 20657669274942672290472]
rank (lower bound): ≥ 11
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 41735582127674531636546296520307
discriminant (Δ): -246955613352378768459234899167754471494203
Faltings height: 6.6214
naive height: 102.7647
primes of bad reduction: 3, 7, 11, 37, 292343, 1492747
regulator: 1611679925.24381677181131277454706745473247293442453774403182
submitted by: jesper-petersen
submitted at: 2026-10-23 21:29:43 UTC
last updated: 2026-10-23 21:29:43 UTC

Witness: 11 independent points log in to add more points →

(2896894, 633545643)
(2924628, 1051815964)
(2928772, 1101273827)
(3292674, 3433489636)
(16777761/4, 56503353749/8)
(7177038, 18595133419)
(8393154, 23819040283)
(13717158, 50567943604)
(28108234, 148941741371)
(239751897/4, 3712094005847/8)
(124803294/25, 1253130272447/125)

Commentary

Specialization t = -169/16 of a generic rank-4 Mordell family obtained from the Burkhardt quartic, with affine parameters (a,b,c,d) = (-10,-32,-9,-4) and generating parameters (u,v,w) = (3,-2,-6). Found using a sixth-power-residue Mestre–Nagao sieve across Burkhardt surface parameters, with 3-isogeny deduplication and conductor-aware candidate selection. PARI/GP 2.17.4 establishes exact rank 11, with bounds [11,11] and 11 independent rational points.

last edited by jesper-petersen at 2026-10-23 21:29:43 UTC · history