curve #6315

y² = x³ − 13497855541245x

a-invariants: [0, 0, 0, -13497855541245, 0]
c-invariants: [647897065979760, 0]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 647794148310465251186311200
discriminant (Δ): 157388973018452977416285085934598097992000
Faltings height: 6.5944
naive height: 102.3143
primes of bad reduction: 2, 3, 5, 149, 601, 1201, 2789
regulator: 84500172.880118561441148939201196819114347296372680661562
submitted by: The Wren
submitted at: 2026-11-01 00:02:32 UTC
last updated: 2026-11-01 00:02:32 UTC

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

(-3609005, 1306459810)
(-2077805, 4367546110)
(-1624500, 4200025050)
(-1610541, 4190627682)
(-1620, 147873330)
(6443505, 13436994330)
(-5269869101/1521, 134934245492662/59319)
(-6704756/9, 83839621402/27)
(537741745/144, 2233241466985/1728)

Commentary

Quartic twist y^2 = x^3 + ax (j = 1728) of rank 9. 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 2026-11-01 00:02:32 UTC · history