curve #6506

y² = x³ + x² − 32762600338616857900x + 71813786852678063732257290000

a-invariants: [0, 1, 0, -32762600338616857900, 71813786852678063732257290000]
c-invariants: [1572604816253609179216, -62047111850149475962191953635264]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 17215711487606289426109405825824671706004755363080
discriminant (Δ): 22767778442359317766029689204653128331191288967673300000000
Faltings height: 10.0382
naive height: 146.4211
primes of bad reduction: 2, 5, 23, 1838988631, 4957400537, 2052598913453611940344986917
regulator: 1243401247499361.3360965767192010687755046253199341981745981184543632551513
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:54:57 UTC
last updated: 2026-11-04 04:54:57 UTC

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

(13928829414/25, 28975461413171562/125)
(1118844827680, -1183447824449635180)
(-1908865481150/289, -36074121857490350/4913)
(8874173264230/1849, -12602579481856246390/79507)
(2554210051090/729, -115869127727426230/19683)
(2856125490, -39222414922590)
(3364010784775/1089, -330869089001057050/35937)
(105576360925344/15625, 778632851536272062172/1953125)
(85662559711489/10000, 647888246548153345513/1000000)
(200503587943950/78961, 1567925191500836745750/22188041)
(7842401405/4, 983573181329315/8)
(427801101550/169, 156899590030519250/2197)
(27927001210/9, 145427567748830/27)
(866891115740/289, -112909061967386620/4913)
(2674512874, -57624926678330)
(-1033461802315/196, 858874782275943385/2744)
(2988911212150/961, -37051499534564050/29791)
(10465929267065/4096, 18128462325027999565/262144)

Commentary

Mestre ten-root construction, roots 0,5,34,43,81,92,99,125,150,169: 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 >= 18: the 18 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 04:54:57 UTC · history