curve #3631

y² + xy + y = x³ + x² − 6893400811406274474520x + 293410482046238687449751456975321

a-invariants: [1, 1, 1, -6893400811406274474520, 293410482046238687449751456975321]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
conductor (N): 3070143408623089579456273867631802
discriminant (Δ): -16226445440889309905780593507495253984060461596572643413958850510848
Faltings height: 11.6033
naive height: 163.0414
primes of bad reduction: 2, 3, 11, 17, 19, 29, 59, 67, 97, 149, 151, 307, 349, 353, 2083, 2297, 3181
regulator: 50860332476.873282815250485315218312563747771104141764972535746
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:23:10 UTC
last updated: 2026-09-29 19:23:10 UTC

Witness: 12 independent points log in to add more points →

(-84867688443, 16345557587117629)
(-5827045413, 18258721035400489)
(16029634949, 13675904580124221)
(51262821509, 8645648983727421)
(65732989595, 11149291936135227)
(105591943477, 27254983549905469)
(225843029509, 101270437519963069)
(26276840364421, 134696871007753207357)
(-1013894745434067/11881, 20890072987733879312641/1295029)
(552627568117/9, 272210522708937559/27)
(2614227240509/25, 3344776913107437513/125)
(2636136431659/25, 3397956436869362117/125)

Commentary

Specialization at T = 121/9 of the rank-9 fibration of X400, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #802 (same T), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:23:10 UTC · history