Hopf surface as an elliptic surface over $mathbb{P}^1$

I saw that the (primary) Hopf surface is an elliptic surface over $mathbb{P}^1$. I want to see why. What I am thinking is by the Hopf fibration $S^1to S^3to S^2$, we can construct a fibration $S^1times S^1to S^3times S^1to S^2$ by taking the product of each fiber and $S^1$. And $S^3times S^1$ is the smooth structure of the Hopf surface, $S^2$ is the smooth structure of $mathbb{P}^1$, $S^1times S^1$ is the smooth structure of the elliptic curve. I wonder if we can show this bundle map is holomorphic so that the fibration $S^1times S^1to S^3times S^1to S^2$ gives the claim that Hopf surface is an elliptic surface over $mathbb{P}^1$?

Do I have to check the local formula to determine if the map $S^3times S^1$ is holomorphic or is there an easier way to see it?

The primary Hopf surface I mean is the compact complex surface given by $mathbb C^2-{0}/mathbb Z$ where $gamma:(z_1,z_2)mapsto (2z_1,2z_2)$ generates $mathbb Z$. And it can be proved that it’s diffeomorphic to $S^3times S^1$.

Attempt: I am thinking maybe I can use the quotient manifold theorem. See the fiber as a complex Lie group, $T:=mathbb C/(mathbb Z+taumathbb Z)$ and it acts on $mathbb C^2-{0}/mathbb Z$ properly and freely, then we get a fiber bundle $mathbb C/(mathbb Z+taumathbb Z)to mathbb C^2-{0}/mathbb Zto (mathbb C^2-{0}/mathbb Z)/T$ and by the long exact sequence of homotopy groups we can see $(mathbb C^2-{0}/mathbb Z)/T$ is $mathbb{P}^1$. And it’s easy to see the action is free, so now the question is if the action is proper? And if so, then by the quotient manifold theorem, we get a holomorphic fiber bundle $Tto mathbb C^2-{0}/mathbb Ztomathbb P^1$