curve #5095

y² = x³ − 705432x

a-invariants: [0, 0, 0, -705432, 0]
c-invariants: [33860736, 0]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 1990537226496
discriminant (Δ): 22467018508184420352
Faltings height: 2.4027
naive height: 52.0133
primes of bad reduction: 2, 3, 7, 13, 17, 19
regulator: 218.1755669272051790496590610311079079337337216
submitted by: The Wren
submitted at: 2026-10-21 02:40:05 UTC
last updated: 2026-10-21 02:40:05 UTC

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

(-608, 14288)
(-399, 14763)
(-152, 10184)
(-399/4, 66633/8)

Commentary

Rational 2-isogeny partner of curve #5094: https://elliptic-rank.icarm.cloud/curve/5094. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (5,7,11,15), 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. 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 02:40:05 UTC · history