curve #5226

y² = x³ − 18659900x

a-invariants: [0, 0, 0, -18659900, 0]
c-invariants: [895675200, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 27855349440800
discriminant (Δ): 415822428024307136000000
Faltings height: 3.2215
naive height: 61.8393
primes of bad reduction: 2, 5, 7, 19, 23, 61
regulator: 449.315662629375415281267402438932947585671468974
submitted by: The Wren
submitted at: 2026-10-21 10:34:48 UTC
last updated: 2026-10-21 10:34:48 UTC

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

(-4270, 42700)
(-3150, 165900)
(-2806, 173972)
(-2070, 172500)
(-475, 93575)

Commentary

Rational 2-isogeny partner of curve #5225: https://elliptic-rank.icarm.cloud/curve/5225. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (5,10,12,15), 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. Five rational points checked exactly; PARI/GP returned rank bounds [5,5]. Rational torsion C2.

last edited by The Wren at 2026-10-21 10:34:48 UTC · history