curve #6522

y² + xy = x³ − x² − 3238475283110794x + 73389709209950130218824

a-invariants: [1, -1, 0, -3238475283110794, 73389709209950130218824]
c-invariants: [155446813589318121, -63408708057886251357132405]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 13621931438642364903907665836794048470758
discriminant (Δ): -153062641903264792249649832467814610524993676388
Faltings height: 7.8216
naive height: 118.8233
primes of bad reduction: 2, 19, 79, 197, 17961409974941, 67494401272556246533
regulator: 2231281655089.847387732681036593542642832193575814982574569877196044547
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:56:50 UTC
last updated: 2026-11-04 04:56:50 UTC

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

(10467820582/81, -981678447281078/729)
(281764196397/16, -149564347892373725/64)
(-245015759316/3721, 7653721482813028/226981)
(143283525369/7225, 79983822220014802/614125)
(228960093286/9025, 74532746567380224/857375)
(28946249966/841, 1266788203839666/24389)
(820884878017/17161, 375654880484247063/2248091)
(5540921154757/85264, 9227139614392846301/24897088)
(1604449485150/22801, 1516095292523466568/3442951)
(1434972180778/25281, -1082021387587939642/4019679)
(759854011/9, -17115517650589/27)
(79099378, -558725794230)
(209496169774451/5067001, -1150519385961527824220/11405819251)
(26924165925749/2053489, -536055575442114967762/2942649737)
(19919838312443/458329, -37673471125679482300/310288733)
(28588210186806/654481, -65448038309523364630/529475129)

Commentary

Mestre ten-root construction, roots 0,6,61,85,95,116,131,146,151,195: 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 >= 16: the 16 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 04:56:50 UTC · history