Galois action is equal to the full group of automorphisms of the lattice, ie, the Weyl group of
E8. In particular, we give explicit elliptic curves over Q (t) whose Mordell-Weil lattices are
isomorphic to E8 and have maximal Galois action. Our main objects of study are del Pezzo
surfaces of degree 1 over number fields. The geometric Picard group, considered as a lattice
via the negative of the intersection pairing, contains a sublattice isomorphic to E8. We …