curve #6523

y² + xy + y = x³ − x² − 15407562762883377x + 782558724074506051182945

a-invariants: [1, -1, 1, -15407562762883377, 782558724074506051182945]
c-invariants: [739563012618402081, -676130734272339671439255345]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 27660077874604450688063675113319693118614
discriminant (Δ): -30466567624358428849883459592491111693978533756928
Faltings height: 8.2435
naive height: 123.5569
primes of bad reduction: 2, 7, 6717853, 785621021, 374353390621395527956277
regulator: 2295607762761.377377462620158677782821712920738482381294433467225492330
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:56:57 UTC
last updated: 2026-11-04 04:56:57 UTC

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

(340558638295, 198740990547623124)
(83849479663/9, -24278087318260516/27)
(1341872819639/289, 1553854323230113556/4913)
(33698372591/121, -5661617000835188/1331)
(5981057563979/36100, -11432166684188383897/6859000)
(1074853194743/12769, -413378587757349740/1442897)
(60511912175/841, -5257587176872444/24389)
(-40746250402205/820836, -887756177849492530991/743677416)
(-248244182801/3025, -203372848467893292/166375)
(4402465967/4, 290326593151951/8)
(320232441660679/2537649, 3721897258696627850132/4042474857)
(-18252127700441/483025, 384354230321553110508/335702375)
(-593060187343545/4800481, 9410866121376301210364/10517853871)
(95184905951/625, 21920661282675476/15625)
(416099671, 8149303051476)
(-5573111177828249/39262756, 81490604720841320055771/246020429096)

Commentary

Mestre ten-root construction, roots 0,1,17,77,95,113,116,151,165,185: 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); for the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, rational points were found by a point search on the quartic y^2 = r(x), and the curves were screened by conductor and by a GRH explicit-formula upper bound on the rank. Rank >= 16: the 16 listed points come from the point search and are independent (height matrix nonsingular at 57 and 115 digits). Torsion trivial.

last edited by Michael Rubinstein at 2026-11-04 04:56:57 UTC · history