curve #219

y² + xy + y = x³ − 76542275904763760832149x + 4899787800875052237542378473282316

a-invariants: [1, 0, 1, -76542275904763760832149, 4899787800875052237542378473282316]
rank (lower bound): ≥ 17
conductor (N): 7123528866816955131107217283375402978917502984525303156208430
naive height: 169.6900
Faltings height: 12.1823
discriminant (Δ): 18328723162152475758783373844977983009407970299659748341711623258000900
primes of bad reduction: 2, 3, 5, 7, 19139, 4326528036023, 130255667791004904223, 3144998910355074343693
regulator: 37834158862449849.9148476359393670234735188524546107191970282166892065793
submitted by: Edgar Costa
last updated: 2026-07-01 22:35:41

Witness: 17 independent points

(-285032152137, -59664226043498732)
(-45981077007, -91225347150091292)
(-58759497417, 95887962972220708)
(488170460409, -289603777638798278)
(608605557828, 428653020829183768)
(-111700832262, 109799527421601418)
(-9060111282, 74783195378869798)
(114555611787/4, 418086210615852599/8)
(38448438639, -44874226224542888)
(-108076640691, 109132194752635672)
(329936981943, 124747886038034548)
(-231587164254, 101021753503259227)
(-277722412167, -68823086504141492)
(2073419014053, 2959722195325353418)
(-3363564171036/25, -14121414026518139497/125)
(-22244479262337/121, 150338829342059702108/1331)
(7726866796671/25, 12970820812301328716/125)

Commentary

Specialization at T = -6286/9 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. Seventeen independent rational points were obtained by searching the low-height Mestre quartic cover and mapping to the minimal model; rank >= 17 follows from the positive-definiteness of their 17x17 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 21:53:12 · history