# Canonical forms for elliptic fibrations with Mordell-Weil group of rank 1 and zero torsion

Consider an elliptic fibration given by the following Weierstrass model: $$E: y^2 + a_1 x y + a_3 y =x^3 + a_2 x^2 + a_4 x + a_6,\quad a_6=a_2 a_4.$$ ( I work with characteristic zero). With the choice $a_6=a_2 a_4$, we easily identify the existence of two sections at $x=-a_2$. More precisely, we get the following two points on each fiber: $$P: (x,y)=(-a_2, 0) \quad \text{and}\quad -P:(x,y)=(-a_2, a_1 a_2-a_3).$$ These two points correspond to the same generator of the Mordell-Weil group since they are actually opposite of each other for the group law of the elliptic curve with the usual zero element as the point of infinity on each fiber.

I have two questions:

1. Is the example above a case of an elliptic fibration of rank 1 with zero torsion?

2. Is there a reference where I can find canonical forms for Weierstrass models with rank 1 or 2 (and no torsion)?

Thank you!

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