curve #1177

y² + xy = x³ − 228650701750535883866170x − 36224857828904644394970864535395100

a-invariants: [1, 0, 0, -228650701750535883866170, -36224857828904644394970864535395100]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/6ℤ
conductor (N): 138952056675401970
discriminant (Δ): 198175867991574047099960189533269983628883789627406319353856000000000000
Faltings height: 12.4049
naive height: 172.9731
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 47, 59, 73, 347, 3467
regulator: 85663.5297982189326679822086603706427880721874797
submitted by: David Renshaw
submitted at: 2026-09-23 21:38:27 UTC
last updated: 2026-09-23 21:38:27 UTC

Witness: 5 independent points log in to add more points →

(-188273633260, -12260101833183370)
(-302322033160, -72590985215112970)
(-780084462475402060/3575881, -386901269091276775587352270/6761990971)
(-8734244133188131660/39702601, -14650410569124913791792984370/250166088901)
(-8633487293135560/36481, -474254713514290896305270/6967871)

Commentary

Obtained by computing a rational isogeny from curve#1088 and mapping all 5 independent witness points to the global minimal model. Isogenies preserve the Mordell-Weil rank; the image points are independently certified by the site verifier. Torsion computed exactly with PARI/GP elltors. Conductor is unchanged by the isogeny.

last edited by David Renshaw at 2026-09-23 21:38:27 UTC · history