curve #220

y² + xy = x³ − 13842353204493057889413x + 635570255797456549699738037418817

a-invariants: [1, 0, 0, -13842353204493057889413, 635570255797456549699738037418817]
rank (lower bound): ≥ 15
conductor (N): 12223008395520638435655400939715776894443327750088254536450
naive height: 164.5873
Faltings height: 11.5945
discriminant (Δ): -4756217026864990828082229613662203105165787694114341605223424000000
primes of bad reduction: 2, 5, 19, 2887, 20611660717, 216219438825740695335021821292755331032329
regulator: 2299012980980795.1736773300246733262723740839069277093718024770082457
submitted by: Edgar Costa
last updated: 2026-07-01 22:42:38

Witness: 15 independent points

(2514318013738/9, 3679561913702325293/27)
(35031218931168258/170569, 5660014854257299777589923/70444997)
(2920373190306/25, 3093840971560740139/125)
(268737294840178/3249, 1392139062719172942527/185193)
(77950760643761618/1129969, 3593459039902576022864833/1201157047)
(1079868038566/25, 1359257684239148771/125)
(5495981224266002/229441, 1959028068119665673087321/109902239)
(-2334203386631278/44521, 327738821395223206667629/9393931)
(-105872842798, 30238515461284999)
(323582351918322, -5820726913620344706361)
(16001135916564622/961, 2024022591810443849120209/29791)
(1116391929791206802/78961, 1179532702855065438160211519/22188041)
(-70392120718, 35512938899164359)
(277081493813293058/316969, 144594497776810957638668973/178453547)
(2513014308877942346562/6869641, -125977339136208237332006595248701/18005329061)

Commentary

Specialization at T = -281 of the Mestre/Fermigier family y^2 = r(x,T) on the sextuple (a) = (1608, -870, -1080, -642, 1542, -558) [r = degree-<=4 remainder in p6(x-T)p6(x+T) = g^2-r, g monic degree 6, p6(x)=prod(x-a_i)]. Located by a Mestre-Nagao sieve of this family. Fifteen independent rational points were obtained from the low-height Mestre quartic cover; rank >= 15 follows from the positive-definiteness of their 15x15 Neron-Tate height-pairing matrix, verified independently in Sage and Magma. Conductor by Tate's algorithm. Proven lower bound on the rank (no exact-rank/Selmer/BSD claim).

last edited by Edgar Costa at 2026-06-25 22:06:54 · history