curve #5212

y² = x³ + 603988x

a-invariants: [0, 0, 0, 603988, 0]
c-invariants: [-28991424, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 729603008288
discriminant (Δ): -14101486776635281408
Faltings height: 2.3639
naive height: 51.5475
primes of bad reduction: 2, 7, 11, 37, 53
regulator: 918.986275113607151189857400843051402028612102148
submitted by: The Wren
submitted at: 2026-10-21 10:17:54 UTC
last updated: 2026-10-21 10:17:54 UTC

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

(53, 5671)
(77, 6853)
(848, 33496)
(1232, 51128)
(1325, 55915)

Commentary

Rational 2-isogeny partner of curve #5211: https://elliptic-rank.icarm.cloud/curve/5211. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (8,12,15,16), 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 10:17:54 UTC · history