Trinomials ax 7 + bx + c and ax 8 + bx + c with Galois Groups of Order 168 and 8 · 168

Abstract

We obtain the curves of genus 2 parametrizing trinomials ax 7 + bx + c whose Galois group is contained in the simple group G 168 of order 168, and trinomials ax 8 + bx + c whose Galois group is contained in G 1344 = (Z/2)3 ⋊ G 168. In the degree-7 case, we find rational points of small height on this curve over Q and recover four inequivalent trinomials: the known x 7 − 7x + 3 (Trinks-Matzat) and x 7 − 154x + 99 (Erbach-Fischer-McKay), and two new examples, 372 x 7 − 28x + 9 and 4992 x 7 − 23956x + 34113. We prove that there are no further rational points, and thus that every trinomial ax 7 + bx + c with Galois group ⊆ G 168 over Q is equivalent to one of those four examples. In the degree-8 case, we again find some rational points of small height and compute the associated trinomials. This time all our examples are new: x 8 + 16x + 28, x 8 + 576x + 1008, and 19453x 8 + 19x + 2, each with Galois group G 1344; and x 8 + 324i + 567, with Galois group G 168 acting transitively on the eight roots. We conjecture, but do not prove, that there are no further rational points, and thus that every trinomial ax 8 + bx + c with Galois group ⊆ G 1344 over Q is equivalent to one of those four examples.