curve #6297

y² = x³ − 25792915457x

a-invariants: [0, 0, 0, -25792915457, 0]
c-invariants: [1238059941936, 0]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 21288783608702384603168
discriminant (Δ): 1098199591603255951087621436735552
Faltings height: 5.0294
naive height: 83.5337
primes of bad reduction: 2, 17, 73, 113, 193, 953
regulator: 105251399.045924234909355025985979879807789074549793291
submitted by: The Wren
submitted at: 2026-10-31 23:51:28 UTC
last updated: 2026-10-31 23:51:28 UTC

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

(-120409, 36877772)
(-3481, 9473276)
(-1921, 7038544)
(-484, 3533222)
(-27064969/169, 10609467848/2197)
(300931769/1681, 2306309566436/68921)
(1654073497/9216, 30031286495293/884736)
(109019021315481/632975281, 411205931938523619252/15925025094679)

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 23:51:28 UTC · history