curve #6072

y² + xy + y = x³ − x² − 7363656800897009034x + 9266861635276348502900472777

a-invariants: [1, -1, 1, -7363656800897009034, 9266861635276348502900472777]
c-invariants: [353455526443056433617, -8006568451288215237512254528281]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 1020148663534491085865258736935914804554554
discriminant (Δ): -11543780925876849096687974021704683194625115019429619900416
Faltings height: 9.8581
naive height: 142.3156
primes of bad reduction: 2, 7, 11, 103, 109, 739, 798425838518175588601339206219617
regulator: 344738918699780175.9647790608215740166741605965160235153720205141847755
submitted by: Michael Rubinstein
submitted at: 2026-10-28 21:14:37 UTC
last updated: 2026-10-28 21:14:37 UTC

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

(-3200127993, 7720005648039)
(-1506744623, 130159428485239)
(388161201, 80417836692087)
(67691977137, 17597955786904023)
(7103557195057, 18932750391507478903)
(-22649953662551/10201, 124792732443855745403/1030301)
(15293971003/9, 1100296546419503/27)
(19392927865/9, 1575379415016301/27)
(178697012481/49, 60307856626730609/343)
(7367356222297/14161, 125842534015437364497/1685159)
(8445528752455/2209, 19970182429736924169/103823)
(51041794437979/4489, 355303468820489595013/300763)
(78979085317969/3721, 696469095889782303747/226981)
(112014933222459/121801, 2431769960486981752667/42508549)
(3483187053358343/822649, 173380151438311457573429/746142643)
(7035718989135505/2024929, 461282819695437823011417/2881473967)

Commentary

Mestre ten-root construction, roots 0,2,22,36,37,47,82,84,97,105: 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 21:14:37 UTC · history