curve #6303

y² = x³ − 86707170872x

a-invariants: [0, 0, 0, -86707170872, 0]
c-invariants: [4161944201856, 0]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 30072533922504820961536
discriminant (Δ): 41720069397962270677690708665270272
Faltings height: 5.3325
naive height: 87.1710
primes of bad reduction: 2, 7, 17, 31, 73, 167, 241
regulator: 572846037.168412003390782031983381018145947676764831305
submitted by: The Wren
submitted at: 2026-10-31 23:55:05 UTC
last updated: 2026-10-31 23:55:05 UTC

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

(-1071, 9636501)
(-4261331868521679/47154556801, 862667156964632264582757/10239659163893951)
(-4088/9, 169443512/27)
(-1687/16, 193510709/64)
(3908529/4, 7367948841/8)
(10995625/36, 9685386325/216)
(5943330649/17424, 231269342328611/2299968)
(59415227936/60025, 13826935347088816/14706125)

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 23:55:05 UTC · history