curve #5101

y² = x³ − 2163777x

a-invariants: [0, 0, 0, -2163777, 0]
c-invariants: [103861296, 0]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 149821788983328
discriminant (Δ): 648361882201957019712
Faltings height: 2.6829
naive height: 55.3757
primes of bad reduction: 2, 3, 7, 11, 17, 19, 29
regulator: 195.6184165092622517504150871774905791737940326
submitted by: The Wren
submitted at: 2026-10-21 02:47:52 UTC
last updated: 2026-10-21 02:47:52 UTC

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

(-1463, 5852)
(-825, 34980)
(1596, 24738)
(3591, 196308)

Commentary

Rational 2-isogeny partner of curve #5100: https://elliptic-rank.icarm.cloud/curve/5100. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (4,8,12,20), 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 02:47:52 UTC · history