curve #1207

y² + xy + y = x³ − x² − 5717390935766x + 5261890646229677772

a-invariants: [1, -1, 1, -5717390935766, 5261890646229677772]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
conductor (N): 695204037449814249
discriminant (Δ): 178828150570193877479607274715069217
Faltings height: 5.9318
naive height: 99.7372
primes of bad reduction: 3, 7, 19, 43, 59, 193, 199, 709, 1201
regulator: 28093356.1816485056207788912370612358517506153078567197
submitted by: David Renshaw
submitted at: 2026-09-24 11:36:44 UTC
last updated: 2026-09-24 11:36:44 UTC

Witness: 8 independent points log in to add more points →

(8051966052/121, 722020623269941/1331)
(7732730492/5329, 55999233557748/389017)
(4623587, 8810634006)
(69077908/9, -549164929588/27)
(678371948, 17668138266207)
(169679052/121, 57679495981/1331)
(96165084/25, -792636526733/125)
(31677603, 177780924462)

Commentary

Obtained by a rational 2-isogeny from curve#1205, mapping its eight independent witness points to the global minimal model. The image curve has torsion Z/2Z. Isogenies preserve the Mordell-Weil rank, and all eight image points are independently certified by the site verifier. The conductor is 695204037449814249, unchanged by the isogeny.

last edited by David Renshaw at 2026-09-24 11:36:44 UTC · history