curve #6135

y² + xy + y = x³ − 21376160230510453x + 1201767174120414373324292

a-invariants: [1, 0, 1, -21376160230510453, 1201767174120414373324292]
c-invariants: [1026055691064501721, -1038326839979121555148941085]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 110588158636195104405800630734472822633358
discriminant (Δ): 1214602353351415357930810588628618740883951116812
Faltings height: 8.1405
naive height: 124.4168
primes of bad reduction: 2, 3, 31, 3241316820388679, 183431619188777602051957
regulator: 17570571023746592.41945437728140375101951311670366780645917423710171186
submitted by: Michael Rubinstein
submitted at: 2026-10-29 12:56:33 UTC
last updated: 2026-10-29 12:56:33 UTC

Witness: 16 independent points log in to add more points →

(52965829, 467039201948)
(66046672, 279339976331)
(71635555, 195112499228)
(198089965, 2177133630686)
(236743753, 3067456410656)
(-51678049283/361, 7905457690543754/6859)
(-18617130425/24964, 4352542570395080947/3944312)
(808942726/49, 316853228263034/343)
(10795370881/49, 919427403421736/343)
(49091531251/625, 1338177868099676/15625)
(85729265488/841, 6982332518635639/24389)
(142210080379/1369, 16188370014606164/50653)
(226236181009/2209, 30573700441588810/103823)
(1260654416473/16384, 234731894310971677/2097152)
(15456987590539/177241, 2151148467668352092/74618461)
(45012800156353/552049, 12283056930464443432/410172407)

Commentary

Mestre ten-root construction, roots 0,4,12,13,15,19,32,62,81,102: with p(x) = prod (x - a_i), write p = q^2 - r with q a monic quintic and deg r <= 4; the genus-one curve y^2 = r(x) carries the ten points (a_i, q(a_i)), and the curve submitted is the minimal Weierstrass model of its Jacobian. Found by enumerating integer ten-root sets 0 = a_1 < ... < a_10 and scoring each by S = sum of -a_p log(p)/p over the good primes 5 <= p <= 199 (with early abort on low partial sums); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 16, proved: 16 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-29 12:56:33 UTC · history