I would like to know how to prove (or even better to see a full proof) of the following "fact".
Let $C_1$ and $C_2$ be two smooth curves and let $\phi : C_1 \rightarrow C_2$ be an isomorphism. Then $ \text{genus}(C_1) = \text{genus}(C_2) $
I am not completely sure this is true since I haven't seen this result explicitly stated, but I Imagine it has to be true.
The motivation for this comes from an exercise from Silverman's book The Arithmetic of Elliptic Curves. I was doing the following exercise and I found that I needed the above mentioned fact in order for my argument for $(i) \implies (ii)$ to work.
2.5 Let $C$ be a smooth curve. Prove that the following are equivalent (over $\bar{K}$):
(i) $C$ is isomorphic to $\mathbb{P}^1$.
(ii) $C$ has genus $0$.
(iii) There exist distinct points $P, Q \in C$ satisfying $(P) \sim (Q)$
I've thought about it but unfortunately I don't really see how to easily relate the dimensions of the Riemann Roch spaces associated to each curve.
I would really appreciate some help with this.
Thank you.