I'm reading Luca De Feo's slides; on the slide 12 it's written that the multiplication map $[m] : E \rightarrow E$ is an isogeny. I do not understand why this isogeny is non-cyclic. At the first glance, there would be a point of order $m$, which generates all other points in the kernel.

Furthermore, I do not understand why the degree of such an isogeny would be $m^2$. Does it mean that such isogeny is inseparable? Let's, for instance, take the curve $E: y^2 = x^3 + x + 4$ over $\mathbb{F}_5$. There are $9$ points on this curve. Let's take the map $P \mapsto 4P$. Obviously, $E[4] = \{\mathcal{O}\}$. What is the degree of an isogeny in this case?

I'm reading Luca De Feo's slides; on the slide 12 it's written that the multiplication map $[m] : E \rightarrow E$ is an isogeny. I do not understand why this isogeny is non-cyclic. At the first glance, there would be a point of order $m$, which generates all other points in the kernel.

Furthermore, I do not understand the following: if the degree of such an isogeny would be $m^2$, does it mean that such isogeny is inseparable? Let's, for instance, take the curve $E: y^2 = x^3 + x + 4$ over $\mathbb{F}_5$. There are $9$ points on this curve. Let's take the map $P \mapsto 4P$. Obviously, $E[4] = \{\mathcal{O}\}$. What is the degree of an isogeny in this case?

I'm reading Luca De Feo's slides; on the slide 12 it's written that non-cyclic isogeny is the multiplication map $[m] : E \rightarrow E$. I do not understand why this isogeny would be non-cyclic. At the first glance, there would be a point of order $m$, which generates all other points in the kernel.

Furthermore, I do not understand why the degree of such an isogeny would be $m^2$. Does it mean that such isogeny is inseparable? Let's, for instance, take the curve $E: y^2 = x^3 + x + 4$ over $\mathbb{F}_5$. There are $9$ points on this curve. Let's take the map $P \mapsto 4P$. Obviously, $E[4] = \{\mathcal{O}\}$. What is the degree of an isogeny in this case?

I'm reading Luca De Feo's slides; on the slide 12 it's written that the multiplication map $[m] : E \rightarrow E$ is an isogeny. I do not understand why this isogeny is non-cyclic. At the first glance, there would be a point of order $m$, which generates all other points in the kernel.

Furthermore, I do not understand why the degree of such an isogeny would be $m^2$. Does it mean that such isogeny is inseparable? Let's, for instance, take the curve $E: y^2 = x^3 + x + 4$ over $\mathbb{F}_5$. There are $9$ points on this curve. Let's take the map $P \mapsto 4P$. Obviously, $E[4] = \{\mathcal{O}\}$. What is the degree of an isogeny in this case?