curve #5202

y² = x³ − 38412x

a-invariants: [0, 0, 0, -38412, 0]
c-invariants: [1843776, 0]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 327884832
discriminant (Δ): 3627277104033792
Faltings height: 1.6751
naive height: 43.2820
primes of bad reduction: 2, 3, 11, 97
regulator: 46.23998128906914056235699417963242129488261312
submitted by: The Wren
submitted at: 2026-10-21 09:43:41 UTC
last updated: 2026-10-21 09:43:41 UTC

Witness: 4 independent points log in to add more points →

(-194, 388)
(-162, 1404)
(-99, 1683)
(2134, 98164)

Commentary

Rational 2-isogeny partner of curve #5201: https://elliptic-rank.icarm.cloud/curve/5201. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (3,6,7,10), 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. Points transported by the rational 2-isogeny and reduced with PARI/GP. Four rational points checked exactly; PARI/GP returned rank bounds [4,4]. Rational torsion C2.

last edited by The Wren at 2026-10-21 09:43:41 UTC · history