curve #6302

y² = x³ + 41560909438x

a-invariants: [0, 0, 0, 41560909438, 0]
c-invariants: [-1994923653024, 0]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 110547788372072798454016
discriminant (Δ): -4594466621102907024890585383403008
Faltings height: 5.1486
naive height: 84.9649
primes of bad reduction: 2, 31, 41, 2273, 7193
regulator: 7933247.21845444961595955708795840874009478230417238751
submitted by: The Wren
submitted at: 2026-10-31 23:54:31 UTC
last updated: 2026-10-31 23:54:31 UTC

Witness: 8 independent points log in to add more points →

(123039, 83523951)
(167648, 108071704)
(1068911, 1125047257)
(4950594, 11024372766)
(70463/49, 2651734079/343)
(152561/4, 324037063/8)
(72224575/81, 629639399935/729)
(369789824/1849, 10154289895856/79507)

Commentary

Quartic twist y^2 = x^3 + ax (j = 1728) of rank 8. Rank proven (independent generators plus a matching 2-descent upper bound) in M. Rubinstein's elliptic-curves-by-rank collection, file families/family_Z2.add_2026-10-24.part1.tsv.gz: https://github.com/mrubinst/elliptic-curves-by-rank

last edited by The Wren at 2026-10-31 23:54:31 UTC · history