Degree of an isogeny from its x-coordinate map ($\mathrm{deg}\varphi =\max (\mathrm{deg}A,\mathrm{deg}B)$)

For an isogeny of elliptic curves $\varphi :E_1\to E_2$, write $x_2\circ \varphi =A(x_1)/B(x_1)$ with coprime polynomials $A,B$. Each $x_i:E_i\to {\mathbb{P}}^1$ is a two-to-one finite morphism, and function field degrees multiply in a tower. Consequently $2\mathrm{deg}\varphi =2\max (\mathrm{deg}A,\mathrm{deg}B)$. This computes the total degree, including an inseparable factor. The zero homomorphism is assigned degree zero separately.