Is it the amount of rational points on the curve(That aren't integers)? So if an elliptic curve has a rank of 1, does that mean it has only 1 rational point on the curve?(In both X and Y)

I've been trying to search it up on google but it the explanations are so vague. I'd be grateful if somebody could explain to me what the rank of an elliptic curve is.

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The rational points on an elliptic curve with rational coefficients form an Abelian group. A theorem of Mordell proves that this group is finitely generated. Thus it is isomorphic to $T\times\Bbb Z^r$ where $T$ is a finite group (the torsion subgroup) and $r$ is a non-negative integer, the rank. So $r$ is positive iff the curve has infinitely many rational points.

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  • $\begingroup$ So if an elliptic curve has a rank of 0, does that mean it has limited rational points? Or none $\endgroup$ – Dean Yang Aug 12 '18 at 18:16
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    $\begingroup$ Let me rewrite my last sentence: So $r$ is zero iff the curve has finitely many rational points. $\endgroup$ – Angina Seng Aug 12 '18 at 18:17

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