curve #756

y² + xy = x³ − 39067869954890072549027630905x + 2975251631263889874076174250349627764668025

a-invariants: [1, 0, 0, -39067869954890072549027630905, 2975251631263889874076174250349627764668025]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 449517592123923690953740840316064540448927891518452310
discriminant (Δ): -7846109474485694297022178534337692317123714544255992769068455706697871501726131200000
Faltings height: 15.2011
naive height: 209.1210
primes of bad reduction: 2, 3, 5, 7, 23, 29, 67, 163, 509, 165391, 3490675691035125664582435380215321467
regulator: 287537628159239.414303832188130083426853941555116670674844370787940061179
submitted by: Valery Asiryan
submitted at: 2026-09-18 01:24:38 UTC
last updated: 2026-09-18 01:24:38 UTC

Witness: 17 independent points log in to add more points →

(-227701393782940, -255372841238799728605)
(-224750577347758, -634816433824237959133)
(-190352168431390, -1874751839212055933605)
(-167209073188864, -2198357604020881469173)
(-142251922744078, -2377851246160330981789)
(-85333080776230, -2384881606532678055565)
(-82613165318530, -2374645730195290533205)
(-37954721663614, -2098424550437393821573)
(-21015835294300, -1946025062721869709895)
(19980177992810, -1484131026081166508605)
(107228482905530, -137734136394032970925)
(1145100247631299820/14641, -1117782993191505300403778965/1771561)
(1278667755839110, -45206537696318836953605)
(-77849147231319305590/477481, -738569541494529421412706257455/329939371)
(12469868890099145/64, -850603437354504745447025/512)
(203493177758027528/2809, -107895407849306180905506593/148877)
(217462539869120, -2182486626266532666955)

Commentary

Specialization at T = -469/509 of the rank-11 fibration of X163 described by wgxli for ICARM #718. Independently recovered polynomial sections and rational-point search yield 17 points with an exact independence certificate, proving rank >= 17.

last edited by Valery Asiryan at 2026-09-18 01:24:39 UTC · history