curve #159

y² + xy = x³ − x² − 1321749187079172070x + 247663242328119893241310696

a-invariants: [1, -1, 0, -1321749187079172070, 247663242328119893241310696]
rank (lower bound): ≥ 17
conductor (N): 84815582549450028799125905482004857770488310327228890
naive height: 136.7901 ★ record for rank ≥ 17
Faltings height: 9.4555 ★ record for rank ≥ 17
discriminant (Δ): 121286283045713541182750044839266946611798283767937312700 ★ record for rank ≥ 17
primes of bad reduction: 2, 5, 11, 13, 10541939988843133587887, 5626250600098076816239435529
regulator: 3039850234600066.55872027922921653126582444319920232392965022269415518269
submitted by: Edgar Costa
last updated: 2026-07-01 22:33:42

Witness: 17 independent points

(17949979918713078398541/6617884875625, -2194610111437610055414042537451414/17024674289667203125)
(1069553418, 7581789774136)
(1914845121457/37249, -96411962079340334623/7189057)
(212468732619099/113569, 2516334388237502223941/38272753)
(2552123861358/1681, -2875329490394494594/68921)
(209259772307129/135424, 2169286183762609158459/49836032)
(4700687044159/1225, 9746723920117144932/42875)
(-4095591542092/3481, -2709694728931345656/205379)
(4122774202837/3721, -2717812918567624915/226981)
(-1855427204865/2401, -3343323765033902131/117649)
(-10901289782/9, -228387183419464/27)
(61860660194/25, 13765447649461718/125)
(60291876902082/16129, 446623168719117198098/2048383)
(13915417197647147/4583881, 1527058322512021184486745/9814089221)
(2959589615, 149194293187741)
(1298436411595/484, 1347253441578324783/10648)
(168529858382/49, 65428320462144828/343)

Commentary

Rank ≥ 17. This curve is the specialization at T = 2454 of the Mestre/Fermigier family y² = r(x,T), where r is the degree-≤4 remainder in p₆(x−T)·p₆(x+T) = g(x)² − r (g monic of degree 6), with p₆(x) = ∏ᵢ(x−aᵢ) on the sextuple (a) = (−1146, −2304, −654, 3054, 2880, −1830). It was found by Alexey Pozdnyakov via a Mestre–Nagao sieve of this family for small-conductor, high-rank specializations. The rank lower bound is witnessed by the 17 listed independent rational points, whose 17×17 Néron–Tate height-pairing matrix is positive-definite; the conductor N = 2·5·11·13·10541939988843133587887·5626250600098076816239435529 is by Tate's algorithm. Proven lower bound on the Mordell–Weil rank (no exact-rank, Selmer, or BSD claim). Curve and search by Alexey Pozdnyakov.

last edited by Edgar Costa at 2026-06-25 21:31:45 · history