Suppose I have two isogenous elliptic curves over $\mathbb{Q}$, $E$ and $E'$. Will the minimal models of $E$ and $E'$ still be isogenous?
2026-04-09 00:24:23.1775694263
Isogeny and minimal models of elliptic curves
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If I interpret the question correctly, the answer is yes. Each elliptic curve is isomorphic to its minimal model by definition: the isomorphism is given by the change of variables transformation. Remember that isomorphisms are degree $1$ isogenies and that the composition of isogenies is an isogeny. Therefore the minimal models are isogenous (with unchanged degree).