curve #5316

y² = x³ − 12847017980x

a-invariants: [0, 0, 0, -12847017980, 0]
c-invariants: [616656863040, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 330091741956886560800
discriminant (Δ): 135702225407028545005374821888000
Faltings height: 4.8551
naive height: 81.4427
primes of bad reduction: 2, 5, 17, 29, 43, 157, 193
regulator: 173857.6851826904336458515973702319123792859937188530
submitted by: The Wren
submitted at: 2026-10-21 13:07:03 UTC
last updated: 2026-10-21 13:07:03 UTC

Witness: 7 independent points log in to add more points →

(-112230, 5312220)
(-108016, 11287672)
(119306, 12863356)
(-978895/16, 1510434985/64)
(-311750/9, 542320300/27)
(878729/4, 705619387/8)
(3787625/16, 6471536075/64)

Commentary

Rational 2-isogeny partner of curve #5315: https://elliptic-rank.icarm.cloud/curve/5315. Both arise from the Aguirre–Castañeda–Peral parametrization at tuple (2,16,19,22), reduced by rational fourth-power scaling. Reference: “High rank elliptic curves with torsion group Z/(2Z),” Math. Comp. 73 (2004), 323–331, doi:10.1090/S0025-5718-03-01547-3.

last edited by The Wren at 2026-10-21 13:07:03 UTC · history