curve #6100

y² + xy = x³ − 376611199004264288295x + 2882549262493214037198461362761

a-invariants: [1, 0, 0, -376611199004264288295, 2882549262493214037198461362761]
c-invariants: [18077337552204685838161, -2490522562821252934467777646182745]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 13406169159886136449996482619880669444469797358
discriminant (Δ): -170837438297692389492480085766629110687115306414695147713691648
Faltings height: 10.7206
naive height: 153.7956
primes of bad reduction: 2, 3, 7, 11, 19, 41, 7519095045599, 4954042034786895640115339629
regulator: 25934491367920437015.9624329966909241210536010521313989485511187287986418
submitted by: Michael Rubinstein
submitted at: 2026-10-29 02:28:47 UTC
last updated: 2026-10-29 02:28:47 UTC

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

(-161769047916884870/7733961, 27280569055819966351230689/21508145541)
(-16464155971914/1849, 186969055729203574575/79507)
(293092238054/25, 35052303443953873/125)
(654260371926/25, 413639767639764561/125)
(3346027178820/289, 1338034692867415527/4913)
(5676884921400/529, 3372740150568346173/12167)
(47627829240189/8464, 755564552019828862509/778688)
(490355510821638/52441, 5074573926802837433361/12008989)
(563321157155526/25, 13370076350502270590961/125)
(1707629754913566/73441, 51502921563012291376323/19902511)
(2546876064655230/237169, 31970378849756552198763/115501303)
(3586828053645606/198025, 124730953411489172939961/88121125)
(35219673324408804/1907161, 3929106860208310484289201/2633789341)
(161786850503052390/51529, 65073839623394425783758183/11697083)
(1014912032549652006/28249225, 897418342299498335885250759/150144630875)
(2178066246574532622/227919409, 1360111483337484040958400837/3440899317673)
(49612367760298764190/3142835721, 164446731506288325688754744969/176190513354981)

Commentary

Mestre ten-root construction, roots 0,8,14,30,54,72,101,108,109,130: 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 2026-10-29 02:28:48 UTC · history