curve #6511

y² + xy = x³ + x² − 19180089265078004x + 57435949186268132132832804

a-invariants: [1, 1, 0, -19180089265078004, 57435949186268132132832804]
c-invariants: [920644284723744217, -49624660103840498298195624221]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 657741528725712916243850244735198959476234712957286
discriminant (Δ): -1424668151219894176584179630096440946225524388265481476
Faltings height: 9.0701
naive height: 132.1486
primes of bad reduction: 2, 3, 19, 42461, 135881456693947204440617798032043224696229959
regulator: 5457686906867988.9296734238659049444739599087272892279082832287728452851958
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:55:33 UTC
last updated: 2026-11-04 04:55:33 UTC

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

(384779894806, 238681181716151268)
(30464566831, -5317282184604007)
(518262084741/16, 373094929925221839/64)
(9281672798214/13225, -30022649867026322868/1520875)
(59921869651/100, -16158893981503731/1000)
(45631548950/121, -13563178024663262/1331)
(502583160359/2209, -836088205302534377/103823)
(799630714/169, -16637157238122078/2197)
(-9318903520209/32761, -37444008776920458459/5929741)
(345686865163/81, 203213597382570241/729)
(12506047087/9, 1406538213534083/27)
(2342664186385/31329, 41651781369836260894/5545233)
(514804750, 13564311400452)
(2756492267410182/1394761, 144903538601333946703476/1647212741)
(9667954875/4, 950972170764741/8)
(580148312810, -441884556026130022)
(1106479994029490/12769, 36805614733909046797714/1442897)
(5739620803452119/10609, -434835590048222841177409/1092727)

Commentary

Mestre ten-root construction, roots 0,3,16,115,120,132,141,156,181,196: 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 04:55:33 UTC · history