3 added 631 characters in body

I do not believe that modular symbols will help. They are good to detect that the rank is positive, but since they only produce torsion points in $E$, they are unfortunately of no use to construct points of infinite order. I fear also that they can not give you the fields that you are after.

If you have the modular parametrisation from a model of $X_0(N)$ to $E$, you can do it easily. But this is not what your after, I know.

Here is another idea. Take a prime $p$ and suppose the point $P$ lies in the formal group. If not you could multiply it to lie in it, but you would change the field. Then $P$ is $p$-adically close to $O$. Now compute its image $z=\log_{\hat{E}}(P)$ under the formal logarithm map into the $p$-adic points of the Lie algebra. The formal version of the modular parametrisation from $X_0(N)$ to $E$ is just $q\mapsto\sum \frac{a_n}{n} q^n$, which has a local inverse around $O$. So you will find a value of $q\in p\mathbb{Z}_p$ that maps to $z$. There is a Tate elliptic curve $A$ defined over $\mathbb{Q}_p$ with multiplicative reduction whose $q$-parameter is this $q$. Computing $j(q)$ to sufficiently high precision, it should be possible to guess its minimal polynomial over $\mathbb{Z}$ and then, using this guess one can verify the claim. (But I have not attempted to do this, so I have no idea if it is feasible.)

(edit:) This can be done for $p=\infty$, too. If $P$ is sufficiently close to $O$ in $E(\mathbb{R})$ then one can recover the $\tau$ in the upper half plane on the imaginary axis with largest imaginary part as above. (This corresponds to integrating the modular form along the imaginary axis until we get $P$ for the first time.) Then one can recover the minimal polynomial of $j(\tau)$ by acting on it with the cosets representatives of $\Gamma_0(N)$.

Note also, once that $j(q)$ is known. We also have to check if the corresponding curve has a $N$-isogeny over its field of definition or otherwise one needs to enlarge it.

In any case, your field will have large degree. In fact, I do not see any reason to believe that the degree of this field should be smaller than the degree of the modular parametrisation.

The question is similar to this .

2 added 3 characters in body

I do not believe that modular symbols will help. They are good to detect that the rank is positive, but since they only produce torsion points in $E$, they are unfortunately of no use to construct points of infinite order. I fear also that they can not give you the fields that you are after.

If you have the modular parametrisation from a model of $X_0(N)$ to $E$, you can do it easily. But this is not what your after, I know.

Here is another idea. Take a prime $p$ and suppose the point $P$ lies in the formal group. If not you could multiply it to lie in it, but you would change the field. Then $P$ is $p$-adically close to $O$. Now compute its image $z=\log_{\hat{E}}(P)$ under the formal logarithm map into the $p$-adic points of the Lie algebra. The formal version of the modular parametrisation from $X_0(N)$ to $E$ is just $q\mapsto\sum \frac{a_n}{n} q^n$, which has a local inverse around $O$. So you will find a value of $q\in p\mathbb{Z}_p$ that maps to $z$. There is a Tate elliptic curve $A$ defined over $\mathbb{Q}_p$ with multiplicative reduction whose $q$-parameter is this $q$. Computing $q$ j(q)$to sufficiently high precision, it should be possible to guess its minimal polynomial over$\mathbb{Z}$and then, using this guess one can verify the claim. (But I have not attempted to do this, so I have no idea if it is feasible.) In any case, your field will have large degree. In fact, I do not see any reason to believe that the degree of this field should be smaller than the degree of the modular parametrisation. The question is similar to this . 1 I do not believe that modular symbols will help. They are good to detect that the rank is positive, but since they only produce torsion points in$E$, they are unfortunately of no use to construct points of infinite order. I fear also that they can not give you the fields that you are after. If you have the modular parametrisation from a model of$X_0(N)$to$E$, you can do it easily. But this is not what your after, I know. Here is another idea. Take a prime$p$and suppose the point$P$lies in the formal group. If not you could multiply it to lie in it, but you would change the field. Then$P$is$p$-adically close to$O$. Now compute its image$z=\log_{\hat{E}}(P)$under the formal logarithm map into the$p$-adic points of the Lie algebra. The formal version of the modular parametrisation from$X_0(N)$to$E$is just$q\mapsto\sum \frac{a_n}{n} q^n$, which has a local inverse around$O$. So you will find a value of$q\in p\mathbb{Z}_p$that maps to$z$. There is a Tate elliptic curve$A$defined over$\mathbb{Q}_p$with multiplicative reduction whose$q$-parameter is this$q$. Computing$q$to sufficiently high precision, it should be possible to guess its minimal polynomial over$\mathbb{Z}\$ and then, using this guess one can verify the claim. (But I have not attempted to do this, so I have no idea if it is feasible.)

In any case, your field will have large degree. In fact, I do not see any reason to believe that the degree of this field should be smaller than the degree of the modular parametrisation.

The question is similar to this .