Let $C_0:$ be a projective closure of affine curve $y^{2}=x^4-7$. $C_0$ has singular point $[0:1:0]$ at infinity.
Let another affine curve be
$C_1: v^{2}=u^{4}(1/u^4-7)=1-7u^4$.
To make smooth projective curve from $C_o$, let glue affine parts of $C_0$ and $C_1$. The glueing maps between the two curves are given by
$(x,y)\mapsto (1/x,y/x^{2})$
and
$(v,v)\mapsto (1/u,v/u^{2})$
Gluing these two curves give smooth projective variety $C$ of genus $1$.
What I want to do is to prove $C$ is normalization of $C_0$.
To prove this, it is enough to prove there is birational and integral morphism between $C_0$ and $C'$.
How can I prove there is integral birational morphism between $C_0$ and $C$ ? I'm stucking with making map between $C_0$ and $C_1$.
Related question:
Weierstrass equation of glued two affine curves
PS(Back ground of my knowledge regarding this question). I'm not familiar with general theory of algebraic geometry. Firstly, I read through Silverman's 'The arithmetic of elliptic curves' . I study continuously Hartshone's 'Algebraic geometry', but not read through. I think I'm familiar with classical varieties. I learned definition of schemes and its morphisms, and definition of properties of morphisms, but not familiar with properties(theorems) of them(morphisms) well. So in this question, 'integral part' is not familiar, but 'birational part' is familiar although even 'Birational part', I'm stuck with. Thus it may be good for me to understand 'Birational part' at first.