I have an elliptic curve $E/\mathbb{Q}(t)$, and I want to compute its rank. Does knowing the rank over $\mathbb{F}_p(t)$ for some prime of good reduction give a bound on the rank over $\mathbb{Q}(t)$? I noticed that magma can compute 2 Selmer groups for elliptic curves over $\mathbb{F}_p(t)$, so I was wondering if that would give me some info on the rank over $\mathbb{Q}(t)$.

Or more generally, how would you go about computing the rank of an elliptic curve over $\mathbb{Q}(t)$?

Note: I've read Stewart and Top's paper "On Ranks of Twists of Elliptic Curves and Power-Free Values of Binary Forms", but am wondering about other techniques.

$\mathbf{Q}$-smooth, hence spreads to a $\mathbf{Z}[1/N]$-smooth proper flat elliptic fibration over $\mathbf{P}^1_{\mathbf{Z}[1/N]}$ for sufficiently divisible $N$. By III.9.3 and preservation of "degree" in flat families, the canonical height is preserved by reduction mod $p$ for all $p\nmid N$, so mod-$p$ reduction preserves linear independence. $\endgroup$