# Thread: elliptic curves: points of finite order

1. ## elliptic curves: points of finite order

I was wondering if anyone could explain the method of finding all the points of finite order over C(\mathbb{Q}), with C: y^2=x^3 + px, where p is a prime bigger or equal to 2.

this would explain a lot of the theory to me.
also - do worked examples or solutions to exercises from cassels, or silverman (+tate) exist? missing the ability to apply things i have heard in lectures

thank you.

2. Originally Posted by zverik136
I was wondering if anyone could explain the method of finding all the points of finite order over C(\mathbb{Q}), with C: y^2=x^3 + px, where p is a prime bigger or equal to 2.

this would explain a lot of the theory to me.
also - do worked examples or solutions to exercises from cassels, or silverman (+tate) exist? missing the ability to apply things i have heard in lectures

thank you.
apply Nagell-Lutz theorem: first of all the (non-zero) torsion points have integer coordinates. if $\displaystyle P(a,b)$ has order 2, then $\displaystyle b=0$ and therefore $\displaystyle a^3+pa=0,$ which has no non-zero solution in $\displaystyle \mathbb{Z}.$

so, there is no point of order 2 on the curve. if $\displaystyle P(a,b)$ has order at least 3, then $\displaystyle b^2 \mid 4p^3,$ which gives us $\displaystyle b=\pm 2, \ \pm p, \ \pm 2p.$ using these possible values of $\displaystyle b$ find possible values of $\displaystyle a$ by solving

$\displaystyle b^2=a^3+pa.$ finally accept only those $\displaystyle P(a,b)$ which have finite order. (remember the theorem only gives a necessary condition!)