curve #6316

y² = x³ − 13928342806092x

a-invariants: [0, 0, 0, -13928342806092, 0]
c-invariants: [668560454692416, 0]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 387997466648029537424624928
discriminant (Δ): 172933175147808093075052835219204270764032
Faltings height: 6.6023
naive height: 102.4084
primes of bad reduction: 2, 3, 11, 13, 23, 349, 877, 1153
regulator: 15355289.756499038714542946779751290004463501177480336083
submitted by: The Wren
submitted at: 2026-11-01 00:03:19 UTC
last updated: 2026-11-01 00:03:19 UTC

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

(-2525952, 4366423776)
(-1420848, 4113591768)
(-752466, 3170891724)
(-583418, 2815575268)
(6067086, 11782281012)
(20035392, 88110645168)
(45824832, 309176322768)
(183031654/49, 103595916164/343)
(211035497062/27889, 84335271548898812/4657463)

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 00:03:19 UTC · history