curve #5221

y² = x³ + 2741508x

a-invariants: [0, 0, 0, 2741508, 0]
c-invariants: [-131592384, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 1670192469792
discriminant (Δ): -1318707653032663584768
Faltings height: 2.7420
naive height: 56.0857
primes of bad reduction: 2, 3, 7, 11, 23, 43
regulator: 412.363160009342867869204425019936966952646873879
submitted by: The Wren
submitted at: 2026-10-21 10:28:45 UTC
last updated: 2026-10-21 10:28:45 UTC

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

(154, 20636)
(322, 30268)
(1012, 61732)
(1978, 114724)
(2898, 179676)

Commentary

Rational 2-isogeny partner of curve #5220: https://elliptic-rank.icarm.cloud/curve/5220. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (3,5,9,10), 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 10:28:45 UTC · history