curve #6067

y² + xy = x³ − 16798981347881908750x + 25472307874744880953526459396

a-invariants: [1, 0, 0, -16798981347881908750, 25472307874744880953526459396]
c-invariants: [806351104698331620001, -22008074004989103800894358348145]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 5527800195841822656759390610258034728913162118
discriminant (Δ): 23111032099180805348620775988120207401547603783808399982592
Faltings height: 9.9620
naive height: 144.4172
primes of bad reduction: 2, 3, 43, 67, 169151, 1890528629892876709227099661969414463
regulator: 152468871943541239898.522281345425290425773708643245196752478775938187167
submitted by: Michael Rubinstein
submitted at: 2026-10-28 21:13:38 UTC
last updated: 2026-10-28 21:13:38 UTC

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

(-436218119866321548916/389636875681, 50361689954103731583877018444354/243214844531961329)
(-1267948607638036/273529, 8744475030393375178886/143055667)
(-336052983475300/332929, 39086909599366401551138/192100033)
(69998821638974/10609, 490963159256871608516/1092727)
(319999575413756/112225, 1032985361167764447166/37595375)
(453974529852476/286225, 8128549905356545651694/153130375)
(667990368829772/380689, 8781794279623902784642/234885113)
(911025671123564/635209, 33309601858874142387034/506261573)
(1383924393067628/703921, 3929887655313873559486/590589719)
(2747510517508124/290521, 132171183050533362445030/156590819)
(3697563425833600/870489, 142430541367453808972218/812166237)
(8587408447706540/19918369, 12028816994088408711351982/88895680847)
(15460413538727144/9102289, 1177656160122732514425298/27461605913)
(26740265741980712/7360369, 2222917063003598209611698/19968681097)
(741443731468461308/180123241, 390314449995315755871492394/2417434017461)
(772523629314482684/236083225, 269983945319623981854838522/3627418752125)
(56585941632960702968/15262378681, 224309635516288191694060364654/1885529524629421)

Commentary

Mestre ten-root construction, roots 0,17,27,44,62,86,99,113,117,137: 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 21:13:38 UTC · history