Let $C_{\lambda}$ denote the non-singular cubic curve $y^2z = x(x-z)(x-\lambda z)$ in $\mathbb{C}\mathbb{P}^2$ for $\lambda \neq 0, 1$.
a) Find projective transformations $\sigma, \tau, \nu$ which send $C_{\lambda}$ to $C_{\lambda^{-1}}$, $C_{1-\lambda}$ and $C_{1-\lambda^{-1}}$, respectively, all of them fixing $[0,1,0]$. (In particular, we see that $C_{\lambda}$ and $C_{\mu}$ are isomorphic if $\mu \in \{\lambda,\lambda^{-1}, 1-\lambda, 1-\lambda^{-1}, (1-\lambda)^{-1}, \lambda(\lambda-1)^{-1} \}$.)
b) When $\lambda = -1$, so that $C_{\lambda} = C_{\lambda^{-1}}$, check that $\sigma$ is of order $4$. It is known that every non-singular cubic curve is isomorphic as a Riemann surface to $\mathbb{C}/\Lambda$ for some lattice $\Lambda$ and that every holomorphic map $\phi: \mathbb{C}/\Lambda \to \mathbb{C}/\Lambda$ is of the form $\phi(z+ \Lambda) = az + b+ \Lambda$ for $a,b\in \mathbb{C}$ with $a\Lambda \subseteq \Lambda$. Identify the lattice $\Lambda$ with $C_{-1} \cong \mathbb{C}/\Lambda$, giving brief reasons.
c) Now let $\lambda = e^{i\pi/3}$ so that $\lambda = (1-\lambda)^{-1}$. Find the order of $\mu$ and again identify the lattice $\Lambda$ with $C_{\lambda} \cong \mathbb{C}/\Lambda$, with brief reasons.
So part a) is fine - the maps are $\sigma([x,y,z]) = [x,\sqrt{\lambda}y,z\lambda^{-1}]$, $\tau([x,y,z]) = [x-z, iy, -z]$ and $\nu$ is formed by firstly applying $\sigma$ and then $\tau$. It is straightforward to compute their orders but I have no idea how such things can be useful for identifying the lattices.
Any help appreciated!
Your maps are wrong (should inverse): $\sigma([x,y,z]) = [x,\frac1{\sqrt{\lambda}}y,\lambda z]$, $\tau([x,y,z]) = [x-z, \frac yi, -z]$.
Your question is the same as (Oxford exam 2011 B3 Geometry (Algebraic Curves) Question 5) From PDF attachment of another post Calculating the lattice of the tori of a non-singular projective cubic curve :