Possible Mordell-Weil groups of Weierstrass elliptic curves

Question

I am finding the Mordell-Weil group of the elliptic curve of the form $y^2=x^3+ax+b$ over the rational field. Using Magma, I found that many of them have the form $\mathbb{Z}/2\times \mathbb{Z}/2\times \mathbb{Z}$ or $\mathbb{Z}/2\times \mathbb{Z}/2\times \mathbb{Z}\times \mathbb{Z}$. Can we determine full candidates of the group form? From Elliptic curves with trivial Mordell–Weil group over certain fields., the answer gave full possibilities of $E(\mathbb{F}_3)$ but did not show how to find them. I've read the book like "Rational points on elliptic curves" but I cannot find the answer. Can anyone help?

Answer 1

Even in the case $b=0$, the Mordell-Weil group　over rational field is not completely determined. The Mordell-Weil group consists of two parts, what we call rank part and torsion part $E(K)_{tor}$.

Rank is an extremely difficult object.
In 'Rational Points on Elliptic Curves' by Silverman and Tate, it is stated that the rank of the elliptic curve $E_p: y^2 = x^3 + px (p\equiv 1\mod8:\text{prime})$ over $\Bbb{Q}$ is believed to be either $0$ or $2$.

For torsion part, the more is known by Mazur that $E(\Bbb{Q})_{tor}\cong \Bbb{Z}/N\Bbb{Z}(N=1,2,,,10,12)$ or $\Bbb{Z}/2\Bbb{Z}\times \Bbb{Z}/2N\Bbb{Z}(1\le N\le 4)$.