curve #5190

y² = x³ − 7672x

a-invariants: [0, 0, 0, -7672, 0]
c-invariants: [368256, 0]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 235438336 ☆ record for quartic twists, rank ≥ 4
discriminant (Δ): 28900526620672
Faltings height: 1.2724
naive height: 38.4496
primes of bad reduction: 2, 7, 137
regulator: 102.8182771564712004478702758274278306467296658
submitted by: The Wren
submitted at: 2026-10-21 09:12:55 UTC
last updated: 2026-10-21 09:12:55 UTC

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

(-63, 483)
(-56, 504)
(-7, 231)
(-343/4, 1323/8)

Commentary

Rational 2-isogeny partner of curve #5189: https://elliptic-rank.icarm.cloud/curve/5189. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (1,2,10,13), 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 09:12:55 UTC · history