curve #5086

y² = x³ + 521775251296068x

a-invariants: [0, 0, 0, 521775251296068, 0]
c-invariants: [-25045212062211264, 0]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 544498825730149822227584521248
discriminant (Δ): -9091392372024411523093673288485328873115291648
Faltings height: 7.5081
naive height: 113.2784
primes of bad reduction: 2, 3, 7, 11, 19, 29, 31, 43, 59, 83, 157
regulator: 4251937.00515022356226059052829892751674803117553030968
submitted by: The Wren
submitted at: 2026-10-21 02:25:52 UTC
last updated: 2026-10-21 02:25:52 UTC

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

(754138, 19847403884)
(3127476, 40772904612)
(43263418, 321793303084)
(139105344, 1662621847824)
(166835893/81, 23995505982511/729)
(268688596/49, 18868119276908/343)
(667343572/9, 18038964094732/27)
(691882986/625, 375965063294484/15625)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (3,10,18,27), 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. Selected in a bounded tuple search using a small-prime sieve and height screening, following the search approach described by David Renshaw on https://elliptic-rank.icarm.cloud/curve/4986. Eight rational points checked exactly; PARI/GP returned rank bounds [8,8]. Rational torsion C2.

last edited by The Wren at 2026-10-21 02:25:52 UTC · history