curve #6103

y² = x³ + x² − 182633547178008x + 1144363346933785543588

a-invariants: [0, 1, 0, -182633547178008, 1144363346933785543588]
c-invariants: [8766410264544400, -988729984349252296926400]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 109913484105165265234727983275144245225200
discriminant (Δ): -175861574568264424375564773240230792360320000
Faltings height: 7.2068
naive height: 110.5014
primes of bad reduction: 2, 5, 8807, 9371, 1322831, 1455911, 25277159, 68392676141
regulator: 250648230537432586.7896653874911056900653482662520482446361249663301280
submitted by: Michael Rubinstein
submitted at: 2026-10-29 02:29:10 UTC
last updated: 2026-10-29 02:29:10 UTC

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

(-15792558, 9480154048)
(-6421322, 45302752860)
(3197238, 24354130340)
(14845542, 41290196348)
(55275414, 399920167204)
(5726286038, 433319615290660)
(-1863876979198/120409, 676679249029797280/41781923)
(-843073424907/111556, 1704590782727307815/37259704)
(-4777996982/1681, 2791521063337440/68921)
(-48035483/4, 320574599745/8)
(25519421/4, 123641854237/8)
(104382726/25, 2665171443476/125)
(16656870022/1369, 1362425379664020/50653)
(1441547143648/398161, 5787211693031655690/251239591)
(3606765722182/15129, 6839059693981126420/1860867)
(25654845628854082/1481711049, 3212666617494838446707120/57035503409157)

Commentary

Mestre ten-root construction, roots 0,2,40,44,58,79,84,99,133,137: 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 2026-10-29 02:29:11 UTC · history