Mazur's theorem

Mazur's theorem tells us that the points of finite order on an elliptic curve over the rationals has to have a particular structure. In particular, if E_tors is the subgroup of E(Q) of points of finite order then E_tors has to have one of the following forms:

1. Z_n, a cyclic group of order n where 1≤n≤10 or n = 12

2. Z₂ x Z_2n, the direct product of a cyclic group of order 2 with one of order 2n for 1≤n≤4

There are examples of curves for each one of these possibilities in Exercise 8.12 on p. 238 of Silverman's The Arithmetic of Elliptic Curves.

I was curious what each of these curves looked like, so I decided to graph both the curves and the points of E_tors. Some of the cases were interesting. Others were not.

In any case, here's what I found.