curve #5027

y² = x³ + 27685464x

a-invariants: [0, 0, 0, 27685464, 0]
c-invariants: [-1328902272, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 3065939667581184
discriminant (Δ): -1358111396687853387350016
Faltings height: 3.3201
naive height: 63.0229
primes of bad reduction: 2, 3, 43, 139, 193
regulator: 7287.59517704413386824959437652810781658166500324
submitted by: The Wren
submitted at: 2026-10-21 01:48:17 UTC
last updated: 2026-10-21 01:48:17 UTC

Witness: 5 independent points log in to add more points →

(193, 73147)
(4632, 477096)
(9457, 1052429)
(18528, 2621712)
(18972672/1849, 92872118784/79507)

Commentary

Rational 2-isogeny partner of curve #5001: https://elliptic-rank.icarm.cloud/curve/5001. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (6,8,9,28), 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. Five rational points checked exactly; PARI/GP returned rank bounds [5,5]. Rational torsion C2.

last edited by The Wren at 2026-10-21 01:48:17 UTC · history