curve #5077

y² = x³ + 130009330308x

a-invariants: [0, 0, 0, 130009330308, 0]
c-invariants: [-6240447854784, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 33804851934269294749728
discriminant (Δ): -140638277156334382263667127844999168
Faltings height: 5.4338
naive height: 88.3862
primes of bad reduction: 2, 3, 11, 13, 19, 37, 47, 2293
regulator: 82561.829260471842802137609604693770245238167730071
submitted by: The Wren
submitted at: 2026-10-21 02:14:45 UTC
last updated: 2026-10-21 02:14:45 UTC

Witness: 6 independent points log in to add more points →

(15466, 44882332)
(285714, 245904516)
(958474, 1002563804)
(1967394, 2805503844)
(1967394/49, 24934751556/343)
(42498214356/776161, 58354589081537268/683797841)

Commentary

Rational 2-isogeny partner of curve #5076: https://elliptic-rank.icarm.cloud/curve/5076. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (3,4,31,60), 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. Points transported by the rational 2-isogeny and reduced with PARI/GP. Six rational points checked exactly; PARI/GP returned rank bounds [6,6]. Rational torsion C2.

last edited by The Wren at 2026-10-21 02:14:45 UTC · history