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I am not sure I am understanding correctly what you are asking for, but if (1) $\mathrm{ord}_{s=1}L(E/K,s) = \mathrm{rk}E(K)$ is known or (2) if the Tate-Shafarevich group $III(E/K)$ is finite, there is an algorithm for calculating the Mordell-Weil group $E(K)$ of an elliptic curve $E$ over a number field $K$. Ask if I shall give you more details.

Edit:

For calculating the torsion, use that $E(K)[m] \hookrightarrow \tilde{E}(k_v)$ for $(v,m) = 1$ and $v$ a place of good reduction.

For calculating the rank:

1. $\mathrm{ord}_{s=1}L(E/K,s) = \mathrm{rk}E(K)$: There is an algorithm which computes $L^{(n)}(E/K,1)$ up to an arbitrary precision, so one can check if it does not vanish (one can not determine if it is equal to $0$. Now the algorithm is as follows: Search in parallel for a non-torsion point and calculate $L(E/K,1)$. If you find a point, move on to $L^{(2)}(E/K,1)$; else you will find after a finite time that $L(E/K,1) \neq 0$. Repeat.

2. $III(E/K) = H^1(\mathcal{O}_K, \mathcal{E})$, $\mathcal{E}$ the Néron model of $E/K$, is finite: This is motivated by the analogy with the (finite) class group $H^1(\mathcal{O}_K, \mathbf{G}_m)$. See http://jmilne.org/math/Books/ectext0.pdf p. 126 for the algorithm.

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I am not sure I am understanding correctly what you are asking for, but if (1) $\mathrm{ord}_{s=1}L(E/K,s) = \mathrm{rk}E(K)$ is known or (2) if the Tate-Shafarevich group $III(E/K)$ is finite, there is an algorithm for calculating the Mordell-Weil group $E(K)$ of an elliptic curve $E$ over a number field $K$. Ask if I shall give you more details.

Edit:

For calculating the torsion, use that $E(K)[m] \hookrightarrow \tilde{E}(k_v)$ for $(v,m) = 1$.

For calculating the rank:

1. $\mathrm{ord}_{s=1}L(E/K,s) = \mathrm{rk}E(K)$: There is an algorithm which computes $L^{(n)}(E/K,1)$ up to an arbitrary precision, so one can check if it does not vanish (one can not determine if it is equal to $0$. Now the algorithm is as follows: Search in parallel for a non-torsion point and calculate $L(E/K,1)$. If you find a point, move on to $L^{(2)}(E/K,1)$; else you will find after a finite time that $L(E/K,1) \neq 0$. Repeat.

2. $III(E/K) = H^1(\mathcal{O}_K, \mathcal{E})$, $\mathcal{E}$ the Néron model of $E/K$, is finite: This is motivated by the analogy with the (finite) class group $H^1(\mathcal{O}_K, \mathbf{G}_m)$. See http://jmilne.org/math/Books/ectext0.pdf p. 126 for the algorithm.

2 added 31 characters in body

I am not sure I am understanding correctly what you are asking for, but if (1) $\mathrm{ord}_{s=1}L(E/K,s) = \mathrm{rk}E(K)$ is known or (2) if the Tate-Shafarevich group $III(E/K)$ is finite, there is an algorithm for calculating the Mordell-Weil group $E(K)$ of an elliptic curve $E$ over a number field . $K$. Ask if I shall give you more details.

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