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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.

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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.

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  • $\begingroup$ Thanks @Lord Shark the Unknown. Do you know an effective way to find at least one rational point of an elliptic curve? (Assume that curve has at least one such point). $\endgroup$ – Leo D Aug 7 '18 at 17:26
  • $\begingroup$ @LeoD That's the hard part... $\endgroup$ – Lord Shark the Unknown Aug 7 '18 at 18:16
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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.

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