Original scientific paper

https://doi.org/10.3336/gm.53.2.04

Trinomials ax8+bx+c with Galois groups of order 1344

Szabolcs Tengely ; Institute of Mathematics, University of Debrecen, P.O.Box 12, 4010 Debrecen , Hungary

Abstract

Bruin and Elkies ([7]) obtained the curve of genus 2 parametrizing trinomials ax8 + bx + c whose Galois group is contained in G1344 = (ℤ/2)3 ⋊ G168. They found some rational points of small height and computed the associated trinomials. They conjecture that the only ℚ-rational points of the hyperelliptic curve

Y2 = 2X6 + 28X5 + 196X4 + 784X3 + 1715X2 + 2058X + 2401

are given by (X, Y ) = (0, ± 49), (-1, ± 38), (-3, ± 32), and (-7, ± 196). In this paper we prove that the above points are the only S-integral points with S={2,3,5,7,11,13,17,19}.

Keywords

Trinomials; hyperelliptic curves; S-integral points

Hrčak ID:

214475

URI

https://hrcak.srce.hr/214475

Publication date:

30.12.2018.

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