Testing to see if $\ell$ is of split or nonsplit multiplicative reduction

Question

Suppose an elliptic curve $E/\mathbb{Q}$ has multiplicative reduction at $\ell$. Are there any other ways of seeing if $\ell$ is of split or nonsplit reduction aside from computing $\widetilde{E}_{\textrm{ns}}(\mathbb{F}_{\ell})$ or computing tangent lines at the node?

Answer 1

You could check whether $-c_6$ is a square in $\mathbb{F}_\ell$. If it is, the reduction is split.

See Wikipedia for a definition of $c_6$. This is just a nice shortcut for calculating the slope of the tangent lines, so there is no new theory here.