what are the examples of elliptic curves defined over $\mathbb{Q}$ with supersingular reduction at a prime $p$ and having a $p$-isogeny over $\mathbb{Q}$ ?



Supersingular elliptic curves over $\mathbb{Q}$
In fact, this cannot happen: an elliptic curve over $\mathbb{Q}\_p$ is supersingular if and only if its associated mod $p$ Galois representation is irreducible, but if it is irreducible as a representation of $\mathbb{F}\_p[G\_{\mathbb{Q}\_p}]$ then it is certainly irreducible as a representation of $\mathbb{F}\_p[G\_{\mathbb{Q}}]$ and thus it has no $p$-isogenies.


This argument doesn't work over larger fields: if $K$ is an extension of $\mathbb{Q}\_p$ it's no longer true that $E/K$ is supersingular if and only if $E[p]$ is irreducible as a $\mathbb{F}\_p[G\_{K}]$-module, cf. [this MSE question](https://math.stackexchange.com/questions/107175/does-the-p-torsion-of-an-elliptic-curve-with-good-reduction-over-a-local-field). In particular, one can have an elliptic curve $E / \mathbb{Q}$ with a $p$-isogeny and bad reduction at $p$, and a number field $F / \mathbb{Q}$ such that $E$ has good supersingular reduction at all primes of $F$ above $p$; my previous suggestion to look for points on $X\_0(p)$ gives lots of examples of this.


