curve #6469

y² + xy = x³ − x² − 9196267049179478584x + 10635869323747244132090559964

a-invariants: [1, -1, 0, -9196267049179478584, 10635869323747244132090559964]
c-invariants: [441420818360614972041, -9189391093731225247503476434725]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 154914169134480285882262406943796131306286378249214
discriminant (Δ): 906811467184020911691274751257668898467467582601972540532
Faltings height: 9.7455
naive height: 142.6096
primes of bad reduction: 2, 7, 23, 53, 5711, 6599071, 321677632097, 748761621964991059060147
regulator: 84688185526665918.857788020792408579631969295032586313664862747553442279520
submitted by: Michael Rubinstein
submitted at: 2026-11-04 02:10:24 UTC
last updated: 2026-11-04 02:10:24 UTC

Witness: 18 independent points log in to add more points →

(14876754622/81, 68978300091257330/729)
(159806093669, -63872351278537997)
(-21149460602/9, 3748042502474080/27)
(888068308718/289, -524415728740220640/4913)
(5768569390335/2809, -3018413870105463661/148877)
(27677016788013/21904, -104205060694316974895/3241792)
(8670998552682/5929, -8097278538095059354/456533)
(31924545677076/9025, 128225921393209203614/857375)
(58030405793491/12321, 366464627115206370554/1367631)
(-3465323577, 29846075717690)
(-3161066477, 90109436890215)
(1219595417, 35130376965580)
(1886159245, 661708977987)
(10806659751/4, 593828052014639/8)
(4735078220, 270655815579318)
(12734930115, 1399584006422407)
(16132854561, 2015222861815339)
(339390616187/169, 35884465153894830/2197)

Commentary

Mestre ten-root construction, roots 0,9,24,34,53,74,77,95,111,129: 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 18, proved: 18 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-11-04 02:10:24 UTC · history