Trinomials ax and ax with Galois Groups of Order 168 and 8·168

• Authors: Nils Bruin;Noam D. Elkies
• Venue: ANTS-V Proceedings of the 5th International Symposium on Algorithmic Number Theory
• Year: 2002

We obtain the curves of genus 2 parametrizing trinomials ax7 + bx + c whose Galois group is contained in the simple group G168 of order 168, and trinomials ax8 + bx + c whose Galois group is contained in G1344 = (Z/2)3 驴 G168. In the degree-7 case, we find rational points of small height on this curve over Q and recover four inequivalent trinomials: the known x7 - 7x + 3 (Trinks-Matzat) and x7 - 154x + 99 (Erbach-Fischer-McKay), and two new examples, 372x7 - 28x + 9 and 4992x7 - 23956x + 34113.We prove that there are no further rational points, and thus that every trinomial ax7 + bx + c with Galois group 驴 G168 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: x8 + 16x + 28, x8 + 576x + 1008, and 19453x8 + 19x + 2, each with Galois group G1344; and x8 + 324x + 567, with Galois group G168 acting transitively on the eight roots. We conjecture, but do not prove, that there are no further rational points, and thus that every trinomial ax8 + bx+ c with Galois group 驴 G1344 over Q is equivalent to one of those four examples.