curve #6515

y² + xy = x³ − x² − 51612695287476549x + 4412307341531288770766729

a-invariants: [1, -1, 0, -51612695287476549, 4412307341531288770766729]
c-invariants: [2477409373798874361, -3812233531934691315847519245]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 422185255072396715012517685057928778442049058
discriminant (Δ): 388956742386668659160742468137759436005989144841052
Faltings height: 8.4897
naive height: 127.0612
primes of bad reduction: 2, 3, 11, 47, 1484303, 8670331, 134820667, 2377018785299191673
regulator: 40178341575823.9795606735027207739965759477295087676473977783988808276441
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:56:00 UTC
last updated: 2026-11-04 04:56:00 UTC

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

(9748802407, -962303751395147)
(19169251984, -2653865763418877)
(-2025192590783/12544, -4104797099339218745/1404928)
(119675069332/1681, 72239455222577843/68921)
(2073984897979/13924, -280988587709656385/1643032)
(571181177257/3721, -74127092318831612/226981)
(164599772, -613584580967)
(447199992848/1369, 239777989626726169/50653)
(9780635714/25, 830073718864577/125)
(1556233501/25, -150038578647964/125)
(106013020, -363569948903)
(62040661048/361, -5393091673645385/6859)
(-4148736907/16, 38496153691845/64)
(212754843905620/597529, -2579032931665666518979/461889917)
(9500942109277/26896, -24264894066213290725/4410944)
(7473798148, -645826882945295)
(4217332450429/44944, -5992125157179749095/9528128)

Commentary

Mestre ten-root construction, roots 0,1,28,41,59,61,66,111,165,182: 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 >= 17: the 17 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 04:56:00 UTC · history