On standardized models of isogenous elliptic curves

Abstract:

Let $E,~E^\prime$ be isogenous elliptic curves over ${\mathbb Q}$ given by standardized Weierstrass models. We show that (in the obvious notation) \[ a_1’ =a_1, \qquad a_2’ = a_2, \qquad a_3’ = a_3 \] and, moreover, that there are integers $t,~w$ such that \[ a_4’ = a_4 - 5t~\text {and}~ a_6’ = a_6 - b_2 t - 7w, \] where $b_2=a_1^2+4 a_2$.