In the book "The Arithmetic of Elliptic Curves", 2nd Edition, by Silverman, there is a theorem called Silverman's Specialization Theorem (Theorem 20.3 pp. 457) which states that:
Let $K$ be a number field, let $C/K$ be a curve and let $E$ be an elliptic curve defined over the function field $K(C)$. Assume that $E$ is nonconstant, i.e., $j(E)\not\in K$. Then the specialization map $$\sigma_{t}: E(K(C))\longrightarrow E_t$$ is well-defined and injective for all but finitely many points $t\in C(K)$.
My question is, is there a way to conclude the injectivity of the specialization map without satisfying the assumption that the $j$- invariant is not in $K$?
I am working with elliptic curves $y^2 = x^3 + a(t)$ over the field $\mathbb{Q}(t)$ whose $j$-invariant is $0$ and I want to use the injectivity of the specialization map.