please explain how to show that $\displaystyle E(\mathbb{Q})[4] $ is a group of order at most 8.
Use Mazur's Theorem: the torsion group of non-singular elliptic curve over $\displaystyle \mathbb{Q}$ has one of the following forms
(i) A cyclic group of orderN with 1 ≤ N ≤ 10 or N = 12.
(ii) The product of a cyclic group of order two and a cyclic group of order 2N
with 1 ≤ N ≤ 4.
Tonio