curve #1208

y² + xy + y = x³ − 97253321847099176x − 5811166088712528801719002

a-invariants: [1, 0, 1, -97253321847099176, -5811166088712528801719002]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
conductor (N): 122310700938581997673650
discriminant (Δ): 44281411863251309409344858226580298292754182020250000
Faltings height: 8.7980
naive height: 128.9619
primes of bad reduction: 2, 3, 5, 7, 13, 43, 47, 79, 89, 103, 199, 227, 313, 433
regulator: 190927584.24006345525709277963872129983992782800085442271
submitted by: David Renshaw
submitted at: 2026-09-24 11:40:13 UTC
last updated: 2026-09-24 11:40:13 UTC

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

(-220901547, -2211857309801)
(-173935998, 2417208096286)
(-62448798, 136581105886)
(-12148706363/81, 1694310408175519/729)
(-2073864385323/16129, 4378053471923781997/2048383)
(344461077, 1248963259411)
(-22312125534/121, -3219724929585859/1331)
(-111103999263/1681, -39467743884003169/68921)
(12544848026/25, -1058759830411914/125)

Commentary

Obtained by a rational 2-isogeny from curve#1204, mapping all nine independent witness points to the global minimal model. The image curve has torsion Z/2Z. Isogenies preserve the Mordell-Weil rank, and all nine image points are independently certified by the site verifier. The conductor is 122310700938581997673650, unchanged by the isogeny. PARI/GP ellrank, supplied with these nine points, returns lower and upper bounds [9,9], proving that the rank is exactly 9.

last edited by David Renshaw at 2026-09-24 11:40:13 UTC · history