curve #6186

y² = x³ − 311207933239563412x + 67086737262927137217359140

a-invariants: [0, 0, 0, -311207933239563412, 67086737262927137217359140]
c-invariants: [14937980795499043776, -57962940995169046555798296960]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 64327933600498682703332055234381017134072
discriminant (Δ): -15273520381607007703035523252548575258654550773873408
Faltings height: 8.8643
naive height: 132.4592
primes of bad reduction: 2, 7, 17, 37, 9305684629, 196251325291998358100491057
regulator: 4534766752812435032.987666254354042509655789825980511910555844522662621
submitted by: Michael Rubinstein
submitted at: 2026-10-30 14:41:29 UTC
last updated: 2026-10-30 14:41:29 UTC

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

(1057811054, 30356881800434)
(-317497907065032/1100401, 13305242904728526395266/1154320649)
(-18175547919218/31329, 40129240027317143122/5545233)
(-4194765028318/908209, 7164734868207430547282/865523177)
(-3490950447938/6561, 4813568282214758962/531441)
(22746024656/121, 5193913924247902/1331)
(198106123689/625, 8401696170781613/15625)
(323364149466/625, 104312084900200514/15625)
(800938097566/2809, 184309876486317302/148877)
(21182681760526/112225, 145947384609261785774/37595375)
(29480213742946/93025, 15272695648596127006/28372625)
(61109018633561/131044, 229322393183854381117/47437928)
(383158583419846/3721, 7500003712022750839862/226981)
(780652876209694/55225, 21794838644024764317622/12977875)
(1556553421689320/6869641, 51574598767860667307830/18005329061)
(43870211678086328/154082569, 2389372346900321091161146/1912626928997)

Commentary

Mestre ten-root construction, roots 0,7,28,51,55,60,73,96,99,113: 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 2026-10-30 14:41:29 UTC · history