curve #6514

y² + xy = x³ + x² − 2559387047119766477x + 1581453270579165442169778705

a-invariants: [1, 1, 0, -2559387047119766477, 1581453270579165442169778705]
c-invariants: [122850578261748790921, -1366375626701778278997804732965]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 1243144522086178951340191789390745525222020024679758998
discriminant (Δ): -7458867132517073708041150736344473151332120148078553988
Faltings height: 9.3858
naive height: 138.7794
primes of bad reduction: 2, 3, 109717225376844435481919, 1888406792729165666167018365607
regulator: 24010083143959589.294498412294610995943716928868549998921907291829728142552
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:55:53 UTC
last updated: 2026-11-04 04:55:53 UTC

Witness: 18 independent points log in to add more points →

(179894196409/64, 65956954023968113/512)
(58810105723, -14256725145775833)
(73698579677/49, 11552426835432230/343)
(39418001331/25, -4785844263230661/125)
(7630826188708/6889, -5874658348288821603/571787)
(7810928815789/7225, -5426310653976959758/614125)
(157445938939/169, -5221114957508997/2197)
(10895370195991/11881, -3062192139642568834/1295029)
(2063990254034/2401, -478812976302279739/117649)
(530813334544/1369, 1288782541340945915/50653)
(182300819, 33480252727967)
(-940010167, 56184917731205)
(1953897067/4, 169290596195661/8)
(-1784045067215/1024, -898663151289118695/32768)
(21594051101711461/23049601, 269960205803936910560385/110661134401)
(-1268746909, 52786447129775)
(-409339780841/225, 49630555920193322/3375)
(-1846817112, 3029483903697)

Commentary

Mestre ten-root construction, roots 0,4,21,60,83,85,104,109,147,173: 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 >= 18: the 18 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 04:55:54 UTC · history