# (Reference request) How to show elliptic curve has positive Mordell-Weil rank

I know there must be a lot of ways to show an elliptic curve has positive Mordell-Weil rank if it really does. And I guess that I am supposed to collect them by myself. But since I am not working in this area, I hope there are some people tell me the important methods that I should not miss and the corresponding reference.

Appreciate for any suggestion.

The surest way of proving an elliptic curve over $\Bbb Q$ has positive rank is to write down an element of infinite order. By the Lutz-Nagell theorem each point of finite order on an elliptic curve $y^2=x^3+ax^2+bx+c$ ($a,b,c\in\Bbb Z$) has integer coordinates. So if you can find a point on the curve with non-integer rational coordinates, the curve must have positive rank.
There several methods to show an elliptic curve over $\mathbb Q$ has positive rank (so infinitely many points), even without giving a point of infinite order.
There are also conjectural ways to show an elliptic curve has positive rank: if you compute the so-called root number (which pari, sage or magma does very fast) and it is $-1$, then the parity conjecture asserts your curve has odd rank (so positive).