curve #5081

y² = x³ − 1364482158732x

a-invariants: [0, 0, 0, -1364482158732, 0]
c-invariants: [65495143619136, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 8443589848063214711328
discriminant (Δ): 162586154149433815630207689524531146752
Faltings height: 6.0215
naive height: 95.4390
primes of bad reduction: 2, 3, 7, 11, 17, 41, 233, 433
regulator: 51174202.82506815966031416536945751521736114734894364
submitted by: The Wren
submitted at: 2026-10-21 02:19:45 UTC
last updated: 2026-10-21 02:19:45 UTC

Witness: 7 independent points log in to add more points →

(-607563, 777648663)
(-866, 34375004)
(1444422, 1021121388)
(-102670395946800363/869412650929, 323744008163371049640103047/810660352217170967)
(-884592786/26569, 922686899884596/4330747)
(1683144177/784, 57936800442807/21952)
(33274451364/1681, 6059110000046796/68921)

Commentary

Rational 2-isogeny partner of curve #5080: https://elliptic-rank.icarm.cloud/curve/5080. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (6,18,20,52), 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. Seven rational points checked exactly; PARI/GP returned rank bounds [7,7]. Rational torsion C2.

last edited by The Wren at 2026-10-21 02:19:45 UTC · history