curve #6018

y² + xy = x³ − 205737886201272577552x + 1133550640131103954544980738304

a-invariants: [1, 0, 0, -205737886201272577552, 1133550640131103954544980738304]
c-invariants: [9875418537661083722497, -979387753088086944533354983478401]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 12061362541463435473927866338570877306640433226
discriminant (Δ): 2250505237548991783441477703671291440686180883389281729511424
Faltings height: 10.4609
naive height: 151.9330
primes of bad reduction: 2, 3, 7, 41, 1741321, 144196043, 2354068357, 1692832290971654689
regulator: 55813287788595495.2832570728677409751639432355059458064060864777860911919
submitted by: Michael Rubinstein
submitted at: 2026-10-28 11:27:53 UTC
last updated: 2026-10-28 11:27:53 UTC

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

(-14811064096, 965243016343856)
(-7955332960, 1505591007596912)
(-26828608705/4, 11897559316811401/8)
(2951309264, 743006655131456)
(7934934704, 25348321209440)
(7975368608, 688677042800)
(8587287752, 7727686869296)
(8890479392, 84560558890736)
(9237537824, 145932636713456)
(9741156080, 231868512453872)
(10303655072, 328002680581616)
(132835245344, 48142524899702000)
(-10683550966874624203348/14230526441569, 60914340110946666399187497483922900/53682341425009076753)
(140131519109/16, 3766841276138309/64)
(1612587815840/121, 1159361886522545872/1331)
(9697468578848/961, 8685376714854160400/29791)
(54390705044188832/4355569, 6502380267767888675798800/9090072503)

Commentary

Mestre ten-root construction, roots 0,5,9,21,27,31,46,55,64,80: 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 17, proved: 17 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-28 22:01:16 UTC · history