the $\ell$-th coefficient in the series for multiplication by $\ell$ in the formal group of $\tilde{E}$, the reduced curve over ${{\mathcal{O}}}_K/\id{l}\cong\F_{\ell}$. By [@Serre2 Proposition 10, page 272], if $v_\ell(a_\ell) > 1$, then $\theta_i|_{I_{\id{l}}}$ for $i=1,2$ would both be squares of fundamental characters of level $2$, whose image is not in $\FF_{\ell}$; hence the valuation is $1$, and the same proposition implies that $\theta_i|_{I_{\id{l}}}$ equals the fundamental character of level $1$ for $i=1,2$, and $*$ is unramified at $\ell$. So $\theta_1/\theta_2$ is unramified, hence trivial, and so $\theta_1 = \theta_2$ and $\theta_1^2=\chi_{\ell}$.

The order of $\theta_1$ equals $\ell -1$ and it factors through a cyclic degree $(\ell-1)$ extension of $K$ unramified outside $\ell$, containing $\zeta_{\ell}$ (the $\ell$-th roots of unity). This implies in particular the existence of a quadratic extension of $K$ unramified outside $\ell$. It can be easily verified (using [@PARI2] for example) that there are no such quadratic extensions, hence this case cannot occur.

$\bullet$ [**Irreducible but absolutely reducible case**]{}. In this case, there is a character of $K$ of order ${\ell^2-1}$ or $\frac{\ell^2-1}{2}$ unramified outside $\ell$ (by the same argument as before). This implies the existence of a quadratic extension of $K$ unramified outside $\ell$ which, as we 