I have an elliptic curve $E$ defined over a ring $R$, I want to compute the pairing
$$ H^1(E,\mathcal{O}_E)\times H^0(E, \Omega_E^1){\rightarrow}R. $$
Clearly we have that $H^0(E, \Omega_E^1)=R \langle w_{can} \rangle$ and I see that $ H^1(E,\mathcal{O}_E)$ could be seen as
$$ \Gamma(E,\mathcal{O}_E(P)/\mathcal{O}_E)=R \left \langle \frac{y}{x} \right \rangle $$
where $x$ has a pole in $P$ with order $2$ and $y$ has a pole at $P$ with order $3$ (with $1,x$ free generators of $\mathcal{O}_E(2P)$ and $1,x,y$ free generators of $\mathcal{O}_E(3P)$. How can I for example compute the product
$$ \langle 1,w_{can} \rangle \qquad \text{or}\qquad \langle y/x, w_{can} \rangle \qquad \text{or}\qquad \langle x, w_{can}\otimes w_{can} \rangle ? $$
For me could be usefull also to understand the case where $R$ is the field of complex number $\mathbb{C}$. (In order to create some intuition).
I think that the point is to understand the so called trace isomorphism
$$ tr: H^1(E, \Omega_E^1) \rightarrow R $$ and compute $$ tr([w_{can}]);\qquad \text{or}\qquad tr( [ y/x w_{can} ] ) \qquad \text{or}\qquad (tr\otimes tr) ( [x w_{can}\otimes w_{can}]). $$
Thank you!
