curve #6516

y² + xy = x³ − x² − 11932544235121449x + 499159103693367217887089

a-invariants: [1, -1, 0, -11932544235121449, 499159103693367217887089]
c-invariants: [572762123285829561, -431273463013639721468211885]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 6792655770912057655076591471745629739000288246
discriminant (Δ): 1100410234887753340122407818422792017718046695852
Faltings height: 8.0588
naive height: 122.6677
primes of bad reduction: 2, 3, 164877103000059467, 2288794248468410813834493641
regulator: 175770043061114.791572134034710919373187492370738094298254425579762473616
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:56:07 UTC
last updated: 2026-11-04 04:56:07 UTC

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

(5335052857/64, -148168554696505/512)
(1721727189011/196, -2258995687151090145/2744)
(59559476356/625, -7452358100669321/15625)
(627312504469/3721, -411036603893423672/226981)
(69204017428/1681, 19208798869985735/68921)
(124874228066/3025, -46147041048570801/166375)
(17673507877/169, 1380642111198280/2197)
(3562097846287/16129, 6019309788423564535/2048383)
(403088612668127/1002001, 7820733705962760293985/1003003001)
(21639586498/169, -2273390267513279/2197)
(271126231/4, -336286080781/8)
(70823987, -96495201320)
(-31237201511/5329, -293417531375800937/389017)
(21914990029913/305809, 18570436756904634615/169112377)
(-20679710765/169, 783342731339416/2197)
(40874419399613/597529, 25277381379659991176/461889917)
(17687880271/625, -6704920978591256/15625)

Commentary

Mestre ten-root construction, roots 0,32,50,61,82,110,117,127,143,178: 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 >= 17: the 17 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 04:56:07 UTC · history