curve #6124

y² + xy = x³ − 1689046724019832173935x + 23974282972546408823139108896697

a-invariants: [1, 0, 0, -1689046724019832173935, 23974282972546408823139108896697]
c-invariants: [81074242752951944348881, -20713780488401708587321618003269529]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 19856314773807030314011608433490767063472777910
discriminant (Δ): 60094304286211659254645369557334274584645061565765012800000000000
Faltings height: 11.1626
naive height: 158.2490
primes of bad reduction: 2, 3, 5, 31, 6309830796427981, 3383747803975121801713397527
regulator: 209775667167923164.122586334626158046320205275825644278125173385157476296
submitted by: Michael Rubinstein
submitted at: 2026-10-29 03:14:30 UTC
last updated: 2026-10-29 03:14:30 UTC

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

(-21714821306, 7100177464191613)
(12709641934, 2135445795436333)
(15075091414, 1392001828722013)
(15316127884, 1302890042350183)
(15822104134, 1100400747631933)
(131784563494, 45717644791806493)
(-12262119422876/361, 44534155097857120447/6859)
(-6573451578606/841, 147747409331081268237/24389)
(-32175813794/9, 147803985618473191/27)
(63769487356/9, 94931947391873791/27)
(517169642854/81, 2673581871719335157/729)
(14256509682726/289, 38276643561401498829/4913)
(28976225300266/1849, 92495729429934588031/79507)
(42200796027646/169, 270620525558066602801/2197)
(44114442150886/529, 261816476081400184811/12167)
(1062248560842871/1444, 34567921593075924197951/54872)
(3956143448842204/36481, 232784685956466632816393/6967871)

Commentary

Mestre ten-root construction, roots 0,18,23,42,71,73,84,85,113,137: 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 17, proved: 17 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-29 03:14:30 UTC · history