curve #6305

y² = x³ − 163490432525x

a-invariants: [0, 0, 0, -163490432525, 0]
c-invariants: [7847540761200, 0]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 34213275554830819680800
discriminant (Δ): 279677160927815003140713721901000000
Faltings height: 5.4910
naive height: 89.0737
primes of bad reduction: 2, 5, 7, 23, 41, 61, 109, 149
regulator: 54804008.9933258283500453743725803037305911834649669102
submitted by: The Wren
submitted at: 2026-10-31 23:56:18 UTC
last updated: 2026-10-31 23:56:18 UTC

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

(-1525, 15789850)
(-26158469204/793881, 51744107170136206/707347971)
(-30956436/625, 1395463803762/15625)
(-5452180/121, 113528018050/1331)
(-33005/9, 661090150/27)
(11937025/9, 39279238550/27)
(170380625/169, 2037275209250/2197)
(1311349025/1024, 45058139834575/32768)

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 23:56:18 UTC · history