This is a remark (I am not an expert). But in his course notes, http://perso.ens-lyon.fr/laurent.berger/ihp2010.php, Laurent Berger explains how $(\Phi,\Gamma)$-modules can be given concretely by giving two matrices $P=Mat(\phi)$ and $G=Mat(\gamma)$ satisfying some semilinear commutation relation etc. This is chapter $10$. In $10.2$ are concrete examples.

And here is an answer in this direction by Berger himself: (phi, Gamma) module of ordinary elliptic curve

This is a remark (I am not an expert). But in his course notes, http://perso.ens-lyon.fr/laurent.berger/ihp2010.php, Laurent Berger explains how $(\Phi,\Gamma)$-modules can be given concretely by giving two matrices $P=Mat(\phi)$ and $G=Mat(\gamma)$ satisfying some semilinear commutation relation etc. This is chapter $10$. In $10.2$ are concrete examples.

And here is an answer in this direction by Berger himself: (phi, Gamma) module of ordinary elliptic curve

This is a remark (I am not an expert). But in his course notes, http://perso.ens-lyon.fr/laurent.berger/ihp2010.php, Laurent Berger explains how $(\Phi,\Gamma)$-modules can be given concretely by giving two matrices $P=Mat(\phi)$ and $G=Mat(\gamma)$ satisfying some semilinear commutation relation etc. This is chapter $10$. In $10.2$ are concrete examples.

And here is an answer by Berger himself, it seems: (phi, Gamma) module of ordinary elliptic curve

This is a remark (I am not an expert). But in his course notes, http://perso.ens-lyon.fr/laurent.berger/ihp2010.php, Laurent Berger explains how $(\Phi,\Gamma)$-modules can be given concretely by giving two matrices $P=Mat(\phi)$ and $G=Mat(\gamma)$ satisfying some semilinear commutation relation etc. This is chapter $10$. In $10.2$ are concrete examples.

And here is an answer in this direction by Berger himself: (phi, Gamma) module of ordinary elliptic curve