What are some examples of the norm for a map of elliptic curves? Given the canonical map $\mathbb{C}[x] \to \mathbb{C}[x,y]/(y^2 - x(x-1)(x-2))$, how can I compute the norm of a $\mathbb{C}(x)$-algebra morphism
$$
\text{Frac}\left( \frac{\mathbb{C}[x,y]}{(y^2 - x(x-1)(x-2))} \right) \xrightarrow{\cdot r} \text{Frac}\left( \frac{\mathbb{C}[x,y]}{(y^2 - x(x-1)(x-2))} \right)
$$
for some $r \in \mathbb{C}(x)$? I am trying to understand how to construct example computations of proper pushforward for maps of curves using the formula at the bottom of page 9 of http://www.cmi.ac.in/~asengupta/Intersection_Theory.pdf

Thank you, Mohan. Given
$$f + g\cdot y \in \text{Frac}\left( \frac{\mathbb{C}[x,y]}{(y^2 - x(x-1)(x-2))} \right)$$
multiplying this element by $y$ gives
$$
f\cdot y + g\cdot x(x-1)(x-2)
$$
hence the matrix of the morphism is given by
$$
\begin{bmatrix}
0 & x(x-1)(x-2) \\
1& 0
\end{bmatrix}
$$
giving a norm of $-x(x-1)(x-2)$.
 A: As Mohan points out in the comments, you have a small mistake in your calculation.  Let $A = \mathbb{C}[x,y]/(y^2 - x(x-1)(x-2))$ and $K = \operatorname{Frac}(A)$.  The matrix of the multiplication-by-$y$ map
\begin{align*}
\lambda_y: K &\to K\\
g(x,y) &\mapsto y g(x,y)
\end{align*}
with respect to the basis $1,y$ is
\begin{align*}
\begin{bmatrix}
0 & x(x-1)(x-2) \\
1 & 0
\end{bmatrix}
\end{align*}
and the norm is the determinant of this matrix.  Note that this only gives you the norm of $y$: to compute the norm of a general element $g + hy$ you can either repeat the computation or use the fact that $\lambda: K \to \operatorname{End}(K)$ is a $\mathbb{C}(x)$-algebra homomorphism.
Here's a slightly different argument you could make.  The extension $K/\mathbb{C}(x)$ is of degree $2$, hence is Galois.  Thus an equivalent definition of the norm of $\alpha \in K$ is the product of all Galois conjugates of $\alpha$.  The conjugate of $g + hy$ is simply $g - hy$, so
$$
N_{K/\mathbb{C}(x)}(g + hy) = (g + hy)(g - hy) = g^2 - h^2 y^2 = g^2 - h^2 x(x-1)(x-2) \, .
$$
Compare this to the norm of a quadratic number field: if $K = \mathbb{Q}[\sqrt{D}]$ for $D$ square-free, then for $\alpha = a + b\sqrt{D}$, the norm of $\alpha$ is
\begin{align*}
N(\alpha) = \alpha \sigma(\alpha) = (a + b\sqrt{D})(a - b\sqrt{D}) = a^2 - b^2D
\end{align*}
where $\sigma$ is the nontrivial element of $\operatorname{Gal}(\mathbb{Q}(\sqrt{D})/\mathbb{Q})$.
