curve #6311

y² = x³ − 3860863358882x

a-invariants: [0, 0, 0, -3860863358882, 0]
c-invariants: [185321441226336, 0]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 954001016061286344210555136
discriminant (Δ): 3683267567247218824874362317014816317952
Faltings height: 6.2815
naive height: 98.5593
primes of bad reduction: 2, 17, 19, 43, 257, 433, 1249
regulator: 1714316666.1586061676287335585208207862160578489548598530
submitted by: The Wren
submitted at: 2026-10-31 23:59:54 UTC
last updated: 2026-10-31 23:59:54 UTC

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

(-1451808, 1595364888)
(-6664, 160401052)
(-46878693040433/24295041, 61719632165936983193/119750257089)
(-15443453435528/10055241, 48428253073353147868/31885169211)
(-187785182825/173889, 123695489537180405/72511713)
(17347361/4, 64410751393/8)
(34028050/9, 169587051130/27)
(8916543554/3721, 481930262550146/226981)
(108698352002/54289, 6886007222429842/12649337)

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 2026-10-31 23:59:54 UTC · history