curve #6301

y² = x³ + 38307956318x

a-invariants: [0, 0, 0, 38307956318, 0]
c-invariants: [-1838781903264, 0]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 93919969104754951495936
discriminant (Δ): -3597882073852862248000525042523648
Faltings height: 5.1283
naive height: 84.7204
primes of bad reduction: 2, 41, 103, 1601, 2833
regulator: 202608401.219795628700438998508955309235639605864546479
submitted by: The Wren
submitted at: 2026-10-31 23:53:56 UTC
last updated: 2026-10-31 23:53:56 UTC

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

(1498657/4, 2069905953/8)
(2435121/100, 30777993831/1000)
(2722513/4, 4674203529/8)
(37174626/361, 486787041198/6859)
(245402959/1849, 6849712778121/79507)
(1020517425/73984, 463733391128745/20123648)
(1042821634/21609, 140669432062174/3176523)
(42448532962/9, 8745688978711982/27)

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 23:53:56 UTC · history