curve #6296

y² = x³ + 21676792718x

a-invariants: [0, 0, 0, 21676792718, 0]
c-invariants: [-1040486050464, 0]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 30072533922504820961536
discriminant (Δ): -651876084343160479338917322894848
Faltings height: 4.9859
naive height: 83.0121
primes of bad reduction: 2, 7, 17, 31, 73, 167, 241
regulator: 14281480.1510130416357812694954305516989294434650511960
submitted by: The Wren
submitted at: 2026-10-31 23:50:54 UTC
last updated: 2026-10-31 23:50:54 UTC

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

(8194, 13348026)
(8978, 13976334)
(1287904, 1471108344)
(4097/4, 37696497/8)
(321976/25, 2096546724/125)
(1778098/1089, 213810950206/35937)
(8001441/4, 22694753823/8)
(181721551/6561, 13250208216223/531441)

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 23:50:54 UTC · history