The hyperkähler structure $\bm{\omega}_{\mathcal{C}}$ is diffeomorphism equivalent to the one given by the Gibbons-Hawking ansatz with $V=z$ on ${\rm Nil}^{3}_{b}(\epsilon,\tau)\times(0,\infty)$ as in Section 2.2, where $b=\deg(L)$, $\tau$ is the modulus of $E$ in the upper half-plane, $A=2\pi b=\epsilon^{2}Im(\tau)$, and $\theta=\theta_{b}$.