curve #6472

y² + xy = x³ − x² − 201479233675716099x + 34919292256244938536223369

a-invariants: [1, -1, 0, -201479233675716099, 34919292256244938536223369]
c-invariants: [9671003216434372761, -30170268465876112421342313405]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 217595709279500935445516459956147588999883282238
discriminant (Δ): -3317464183675271261802343948491426141892220521000548
Faltings height: 8.7466
naive height: 131.1533
primes of bad reduction: 2, 3, 7, 11, 7086647963253736811, 7384566114317459889120451
regulator: 1625485584276615.32399401799947593183256931240581403823355457489684751514
submitted by: Michael Rubinstein
submitted at: 2026-11-04 02:10:47 UTC
last updated: 2026-11-04 02:10:47 UTC

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

(2346652699/9, 9036213130705/27)
(759553880904, -661969612832783717)
(245366032501/2601, 542926973543005670/132651)
(212137491375/676, -28323255010852937/17576)
(1576919133968/6889, -509496980082911585/571787)
(158928767164/2025, -403327781762544053/91125)
(1603468271744/3025, 1460445789617400437/166375)
(2249723641813/3249, 2792656779854299249/185193)
(201911530174/225, 81053164117841713/3375)
(-518317363, 319665286145)
(-422281746, 6685884092743)
(142182510, 3024298541447)
(270208878, 454322261767)
(1134172247/4, 6129031303495/8)
(357628001, 2933165098754)
(1601383550, 61796518139363)
(3803084368/9, 135627045145121/27)

Commentary

Mestre ten-root construction, roots 0,6,51,52,83,90,110,114,120,140: 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 2026-11-04 02:10:47 UTC · history