TITLE: Supersingular elliptic curves over $\mathbb{Q}$
QUESTION [9 upvotes]: what are the examples of elliptic curves defined over  $\mathbb{Q}$ with supersingular reduction at a prime $p$ and having a $p$-isogeny over $\mathbb{Q}$ ?

REPLY [17 votes]: I felt that there couldn’t be such an isogeny, but when I saw @DavidLoeffler’s answer, I realized that I had an argument, too. The formal group of the elliptic curve would be of height two, defined over $\mathbb Z_p$, but such things don’t have $p$-isogenies: the quotient formal group is definable only over a suitably ramified extension.