curve #6069

y² + xy = x³ − x² − 230685453317905775709x + 1257046703414879260209833270465

a-invariants: [1, -1, 0, -230685453317905775709, 1257046703414879260209833270465]
c-invariants: [11072901759259477234041, -1086088351700627622904628298128589]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 74796929460358181130135996875568438477322157430
discriminant (Δ): 103038890251472901697685300828111131547570191894990683895312500
Faltings height: 10.6442
naive height: 152.2764
primes of bad reduction: 2, 3, 5, 13, 37, 103, 7146311, 2347345661258664947141613502279799
regulator: 48188017170914733113.7255820717139793444992906872271825056090036032731384
submitted by: Michael Rubinstein
submitted at: 2026-10-28 21:14:02 UTC
last updated: 2026-10-28 21:14:02 UTC

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

(12093342911, 485717153265357)
(236338981411, 114663364254932857)
(-8615036290721641/1181569, 2051537850259205982686571/1284365503)
(-2138209100163539/234256, 182897506273889353191193/113379904)
(-63058526900629/3721, 124219776392148178612/226981)
(391772993131/7396, 709659274886528417867/636056)
(336712180324/81, 443392066413675431/729)
(29826266886689/1849, 104674419525191544249/79507)
(42038739892604/6241, 46239890525613712603/493039)
(71187424629256/121, 600428376402748361117/1331)
(159187658185621/961, 2000280063564330851212/29791)
(2449539032381701/229441, 11567922878090602766873/109902239)
(5139369888788239/3396649, 5976490190948053801855324/6260024107)
(27945611661303791/17956, 4671427856250156162284253/2406104)
(13518858085908214/5968249, 12594440621547341950024099/14580432307)
(26031095399032126/168921, 4180167802907900736568727/69426531)
(29928929423542136/1485961, 3960805981916095256933443/1811386459)

Commentary

Mestre ten-root construction, roots 0,1,21,39,65,66,75,96,98,113: 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 21:14:02 UTC · history