curve #6074

y² = x³ + x² − 640486144557379276x + 196102252580419637122595024

a-invariants: [0, 1, 0, -640486144557379276, 196102252580419637122595024]
c-invariants: [30743334938754205264, -169432346413942576106447332288]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 1558190906638665417508116153288187194306360
discriminant (Δ): 202444598074402612402085312538318584306422115912121600
Faltings height: 9.0617
naive height: 134.6166
primes of bad reduction: 2, 3, 5, 13, 23971, 118730017682438237, 350953342328176603
regulator: 507942562301564194.0032458377865199101955425959373781386982239597971951
submitted by: Michael Rubinstein
submitted at: 2026-10-28 21:15:01 UTC
last updated: 2026-10-28 21:15:01 UTC

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

(-823844858, 12829777849278)
(432399404, 42608837700)
(3390797747, 192380020079328)
(-2680347262748462/3523129, 103092694677618722320374/6612913133)
(-277345190078704/3087049, 86259375957619826757900/5423945093)
(10637862526/25, 97697590753074/125)
(49791215246/121, 1984367424091650/1331)
(108515182934/361, 38026268723517750/6859)
(124952444615/289, 341857268934246/4913)
(260555510690/529, 4311564204985986/12167)
(549495267494/961, 122192776698343650/29791)
(823368292397/1156, 395481110506524837/39304)
(1476270180506/1225, 1469335056120816354/42875)
(6640215145046/18769, 9548206410151514430/2571353)
(383606550908912/73441, 7429787144464736244492/19902511)
(4441687626537206/477481, 294959298348318771438570/329939371)

Commentary

Mestre ten-root construction, roots 0,1,11,30,37,51,101,102,104,129: 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 2026-10-28 21:15:01 UTC · history