curve #6497

y² + xy + y = x³ − x² − 407653364784444593911182x + 100787510356269120398445079814488989

a-invariants: [1, -1, 1, -407653364784444593911182, 100787510356269120398445079814488989]
c-invariants: [19567361509653340507736721, -87080408947728466897463108927433671481]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 2911330511056578480488235124713909835325639289152655130
discriminant (Δ): -52670270074838607435721363192930238258297688792441982494725338880000000
Faltings height: 12.4034
naive height: 174.7198
primes of bad reduction: 2, 5, 7, 37, 43, 6512969149539799, 4013694110277216173007960619324351
regulator: 13804547861455468.57584723328821560715449051374838870820392343068363878612422
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:53:53 UTC
last updated: 2026-11-04 04:53:53 UTC

Witness: 19 independent points log in to add more points →

(198913717017, 166042115293260051)
(48243265631177, -335055873723976631549)
(-613641417703, -346222555694247149)
(367814909547, -24645621780256149)
(2253591710965277/6241, -12752146285961467901111/493039)
(260915446403473/729, -532697368915241181287/19683)
(283657637188363/529, 2320470872889889943117/12167)
(547609739949, 204371469938180763)
(1221819352301433/2209, 21907245380503595901853/103823)
(355811206297, 28038154160000851)
(15469622473803/49, 20451891338141034143/343)
(114773846297, 235608698200592851)
(251364977697, 119163898433528451)
(-445688158453, 440390064382086601)
(561533554045364763/1630729, 73456737268450746093662983/2082440933)
(-32397090263, 337580066186632051)
(377486094777, 26347458426958451)
(9039248838928617/28561, 284367185760379609494459/4826809)
(24015087168833737/83521, 2068726748237132982603219/24137569)

Commentary

Mestre ten-root construction, roots 0,5,37,74,79,81,138,140,141,187: 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); for the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, rational points were found by a point search on the quartic y^2 = r(x), and the curves were screened by conductor and by a GRH explicit-formula upper bound on the rank. Rank >= 19: the 19 listed points come from the point search and are independent (height matrix nonsingular at 57 and 115 digits). Torsion trivial.

last edited by Michael Rubinstein at 2026-11-04 04:53:53 UTC · history