If $C$ is a projective integral curve over $\mathbb C$, suppose its arithmetic genus $p\_a(C)=1$, then what kind of curve $C$ could be?

According to Hartshorn Ex. 1.8 page 298, if $\tilde C \to C$ is the normalization, then $p\_a(\tilde C)+\delta = p\_a(C)=1$, hence either $C$ is an elliptic curve or $\tilde C = \mathbb P^1$ and $\delta =1$. Ex. 1.8 (c) claims that node or cusp will have $\delta=1$.

It seems that these are the only possibilities, but why?