There is an exercise (3.21) in Silverman, Arithmetic of Elliptic Curves as-
Let $C$ be a curve of genus one. For any point $O \in C$, we can associate to the elliptic curve $(C,O)$ it's $j$-invariant $j(C,O)$. We prove in this exercise that the value of $j(C,O)$ is independent of the choice of the base point $O$. Thus we can assign a $j$- invariant to any curve $C$ of genus one.
My question- There is nothing mentioned about $C$ being non singular or an elliptic curve and Silverman has only defined $j$-invariant for elliptic curves. So, I'm confused by the language of this question.
$j$-invariant is defined for only elliptic curves in the book (Chapter 3), as they have a Weierstrass form, we define their $j$- invariant in terms of coefficients of Weierstrass equation.
Can someone help clarify this question for me? Thank you in advance.
"My question- There is nothing mentioned about $C$ being non singular or an elliptic curve and Silverman has only defined $j$-invariant for elliptic curves."
Yes, indeed we need to assume that the curve is non-singular. But Silverman assumes this already implicitly in the definition in a curve of genus $1$, because the genus is only defined for non-singular curves - see the comment above.
The $j$-invariant of an elliptic curve $E$ is defined by $$ j(E)=\frac{c_4(E)^3}{\Delta} $$ from the Weierstrass equation. There we need that $\Delta\neq 0$, i.e., that the curve is non-singular. So we cannot use this definition if $\Delta=0$.