Mordell-Weil rank of some algebraic surface Consider an elliptic curve  $S:y^2 = x^3 + t^2x + (t^3 + 1)$ over $k(t)$, where char($k$) is 0.
How can I calculate the Mordell-Weil rank of the surface, or how to get its Picard number $\rho(S)$ of the Neron-Severi group. I guess $\rho(S) = 2$ and its Mordell-Weil rank is 0, but I cannot prove it.
 A: I assume that $k$ is algebraically closed; if you are interested in non-closed fields then one needs to calculate the Picard group over the algebraic closure first then use Galois theory to try to decide what the Picard group over the ground field is.
Any elliptic surface of the form $y^2 = x^3 + a_4(t)x + a_6(t)$ with $\deg a_i \leq i$ is a rational elliptic surface. So your surface is rational.
Let $S$ be the unique smooth projective relatively minimal surface corresponding to your equation. Then $S$ is the blow-up of $\mathbb{P}^2$ in the base-locus of a pencil of plane cubic curves. It follows that $\rho(S) = 10$.
To calculate the rank of the Mordell-Weil group one then uses the Shioda-Tate formula, which requires calculating the Kodaira symbol of the singular fibres using Tate's algorithm.
Rather than writing out the full details I think this is a good exercise to have a go at yourself. You can read all about these results and this method in the book:
Schutt, Shioda - Elliptic surfaces
Chapter 8 in particular is dedicated solely to rational elliptic surfaces.
