curve #5090

y² = x³ − 43244058779327637x

a-invariants: [0, 0, 0, -43244058779327637, 0]
c-invariants: [2075714821407726576, 0]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 135695137938136502727507141369888
discriminant (Δ): 5175583515579988446147720786121303391752754551286592
Faltings height: 8.6124
naive height: 126.5305
primes of bad reduction: 2, 3, 7, 11, 17, 19, 23, 31, 43, 47, 107, 179
regulator: 1298534027.8731865815221134486102816730846476361031382415
submitted by: The Wren
submitted at: 2026-10-21 02:32:30 UTC
last updated: 2026-10-21 02:32:30 UTC

Witness: 9 independent points log in to add more points →

(-143692500, 1801929814650)
(-18559257, 892291958046)
(-14360409, 786155803902)
(270538800, 2846379295620)
(527366448, 11129386748004)
(702885568, 17800664870296)
(-1465457491977/66049, 16532173097887842954/16974593)
(-128485736211/625, 7020717598041462/15625)
(-128485736211/961, 54865208162403954/29791)

Commentary

Rational 2-isogeny partner of curve #5089: https://elliptic-rank.icarm.cloud/curve/5089. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (9,27,29,41), 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. Nine rational points checked exactly; PARI/GP returned rank bounds [9,9]. Rational torsion C2.

last edited by The Wren at 2026-10-21 02:32:30 UTC · history