curve #6304

y² = x³ − 135760793836x

a-invariants: [0, 0, 0, -135760793836, 0]
c-invariants: [6516518104128, 0]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 36861986285961791189792
discriminant (Δ): 160141200657725378390100251511107584
Faltings height: 5.4446
naive height: 88.5161
primes of bad reduction: 2, 29, 53, 59, 241, 1553
regulator: 38972935.4845099109027135339181055820201672440352620089
submitted by: The Wren
submitted at: 2026-10-31 23:55:44 UTC
last updated: 2026-10-31 23:55:44 UTC

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

(-355475, 57800235)
(-112572, 117712788)
(1795268, 2354229972)
(19427733, 85616026743)
(117715968, 1277176561728)
(-108224406407/304704, 9825968581184717/168196608)
(-11407286619/121801, 4635813546476865/42508549)
(-640381103/1764, 2813194185479/74088)

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 23:55:44 UTC · history