curve #6319

y² = x³ + 103171661828x

a-invariants: [0, 0, 0, 103171661828, 0]
c-invariants: [-4952239767744, 0]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 21288783608702384603168
discriminant (Δ): -70284773862608380869607771951075328
Faltings height: 5.3760
naive height: 87.6926
primes of bad reduction: 2, 17, 73, 113, 193, 953
regulator: 26944358155.7566041367948866524108492307940030847470825
submitted by: The Wren
submitted at: 2026-11-01 00:08:19 UTC
last updated: 2026-11-01 00:08:19 UTC

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

(11294588176/120409, -4281992113223688/41781923)
(25780798096/3481, -4143359289389352/205379)
(13424896, -49202896032)
(25792681201/484, -4142405637057513/10648)
(153664/169, -21279171984/2197)
(2142722150416/61324561, -29003609835316413048/480232637191)
(174378902569/4875264, -657962962789378859/10764582912)
(57307165632553326736/2549655447656769, -6214795759245376436580112900888/128742590285046479100897)

Commentary

Quartic twist y^2 = x^3 + ax (j = 1728) of rank 8: the rational 2-isogenous partner (a -> -4a) of y^2 = x^3 - 25792915457x, from M. Rubinstein's elliptic-curves-by-rank collection (families/family_Z2.add_2026-10-24.part1.tsv.gz). The eight generators were transported through the 2-isogeny (x, y) -> (y^2/x^2, y(a - x^2)/x^2) and checked exactly on this model; rank is preserved under isogeny. https://github.com/mrubinst/elliptic-curves-by-rank

last edited by The Wren at 2026-11-01 00:08:19 UTC · history