(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.   
 A: 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.
A: 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. 
The easiest way for a specific elliptic curve is to use either pari, sage or magma, asking either for the rank, or for the analytic rank; if this analytic rank is zero or 1, then it is equal to the actual rank this is a theorem by Kolyvagin). 
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). 
The bibliography concerning all this is quite bast; without knowing what do you want to know exactly it is difficult to point you to some specific place. 
