I am looking for an elliptic curve $E$ with Weierstraß coefficients in $\mathbb{Q} $ so that for some prime $p$ the following conditions are satisfied:

(1) ${\large Ш}_{p^{\infty}}(\mathbb{Q},E)$ contains an element of order $p^{2}$.

(2) $\operatorname{Gal} (\mathbb{Q}(E_{p})/ \mathbb {Q} ) $ is nonsolvable, so in particular $E_{p}(\mathbb{Q})=0$ and $E$ has no CM.

I need this to verify (or falsify) a calculation of mine, but all curves I've found so far for which the exact structure of $\large Ш$ is known violate at least one of these conditions.

If such a curve exists, a Weierstraß equation or a Cremona label would be nice.