curve #6176

y² + xy + y = x³ − x² − 17647330988340345402x + 28400013493631012572109825145

a-invariants: [1, -1, 1, -17647330988340345402, 28400013493631012572109825145]
c-invariants: [847071887440336579281, -24537611654685371368821374318745]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 4190506510606183610086555829247330835129764106
discriminant (Δ): 3301949366703454340468160175261458814790094726110976950272
Faltings height: 9.8804
naive height: 144.5650
primes of bad reduction: 2, 7, 59, 97, 521, 14083, 3773701, 5545081, 7089637, 48048789943163
regulator: 788195472308523695.768949439230876569887105436989149043139828192830290140
submitted by: Michael Rubinstein
submitted at: 2026-10-30 14:40:19 UTC
last updated: 2026-10-30 14:40:19 UTC

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

(-706103897, 201268362131617)
(2560486745, 1020494782255)
(16955937745, 2145701176816999)
(-7272095770055/1521, 3500198972727312881/59319)
(24516485377/81, 110765621113039855/729)
(101390728249/25, 19176754830707547/125)
(207998312353/81, 2951875900739791/729)
(619670339217/361, 385899286293705845/6859)
(707039513001/625, 1553400532327585799/15625)
(965222372929/289, 402605251136938503/4913)
(1578612518161/729, 362794850113251725/19683)
(8028137751649/3136, 38200376536399755/175616)
(16128680761945/4489, 32078524500597224605/300763)
(24924759490629/6889, 62406149722163978209/571787)
(80358020274297/72361, 1963160479643315294163/19465109)
(146565135739153/9409, 1715537196854497472935/912673)
(3724011707395341/638401, 179583303428908210664333/510082399)

Commentary

Mestre ten-root construction, roots 0,11,55,69,83,93,94,97,108,124: 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 17, proved: 17 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-30 14:40:19 UTC · history