curve #6500

y² = x³ − 35770506256734097807x + 51646522659139630586844252370

a-invariants: [0, 0, 0, -35770506256734097807, 51646522659139630586844252370]
c-invariants: [1716984300323236694736, -44622595577496640827033434047680]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 71282932021622334919443458087017223694230637218579805784
discriminant (Δ): 1776940929435001564871886523193165352249781324584757398583552
Faltings height: 10.2622
naive height: 146.6846
primes of bad reduction: 2, 19, 41, 3769, 87587, 49118132515147, 705424396042938613529302168457
regulator: 97221898382089779.59550479739059723608661250666460109337700891927176513648260
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:54:14 UTC
last updated: 2026-11-04 04:54:14 UTC

Witness: 19 independent points log in to add more points →

(33318295961, -5987217156573782)
(96660952661601/25, -950333606138319520526/125)
(-20558953365959/3481, 48989501846135497322/205379)
(66338265739/2025, 20472672507940336112/91125)
(1243666497737/5329, 80959678279595498758/389017)
(37023883349/25, 5477331643055618/125)
(4329473728933/729, 4342711026829763186/19683)
(138031490330441/14641, 1316652126954688004038/1771561)
(60947953323085/4761, 427266449148921480010/328509)
(4584701863471521/473344, -255149307465193475427343/325660672)
(41946011129/4, -7287053903704811/8)
(191171018337/16, -73807506046667887/64)
(13616284998790/1849, -34435641953511923240/79507)
(349790335017/49, -137231257927682918/343)
(83292107433/16, 5163602889261589/64)
(-26875770258314/4225, -40539375198951218816/274625)
(27448177541, -4444022449590302)
(-1021817659234303/155236, 2669446693321859238953/61162984)
(1195733058977/49, -1270044458594763118/343)

Commentary

Mestre ten-root construction, roots 0,4,59,72,73,82,108,121,138,163: 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 >= 19: the 19 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 04:54:14 UTC · history