curve #6317

y² = x³ − 17719768397452x

a-invariants: [0, 0, 0, -17719768397452, 0]
c-invariants: [850548883077696, 0]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 627980384118677240480184608
discriminant (Δ): 356085342871228953501618765896360858202112
Faltings height: 6.6625
naive height: 103.1307
primes of bad reduction: 2, 11, 97, 881, 1697, 2777
regulator: 2176803427.1504709135016737526704590941302220708360576306
submitted by: The Wren
submitted at: 2026-11-01 00:03:53 UTC
last updated: 2026-11-01 00:03:53 UTC

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

(-2357003, 5354559177)
(-37477115563058/13271449, 253630529263292627052/48347888707)
(-321811911872/2954961, 7054015978840934224/5079577959)
(-41767279664/403225, 346787186470924584/256047875)
(-10764762800/3249, 875286092696840/185193)
(-6889448192/1681, 134026463946144/68921)
(-414551907/169, 11771161918131/2197)
(11628390361/36, 1253841157866563/216)
(10068050225825/872356, 29745391914495103215/814780504)

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 00:03:53 UTC · history