curve #6314

y² = x³ + 10860647005268x

a-invariants: [0, 0, 0, 10860647005268, 0]
c-invariants: [-521311056252864, 0]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 235907306746073553639503648
discriminant (Δ): -81987391505042662209680955414821158965248
Faltings height: 6.5401
naive height: 101.6621
primes of bad reduction: 2, 97, 101, 197, 829, 1697
regulator: 70220731955.687175347267157344786121082832272037794175417
submitted by: The Wren
submitted at: 2026-11-01 00:01:41 UTC
last updated: 2026-11-01 00:01:41 UTC

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

(130896, 1193255208)
(6464/25, 6623972688/125)
(110399588/292681, 10134588245335164/158340421)
(3081027049/484, 192577290047979/10648)
(3872427264/169, 243456201268704/2197)
(3182337183041/407044, 6160748841457322367/259694072)
(4086715470309/1934881, 15310711056191163231/2691419471)
(61477154422052/21836929, 742114704260738151228/102043969217)
(75473227158436/2074164849, 59387364213315736974652/94463689718007)

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 00:01:41 UTC · history