 Sign of a lift of a Legendre parameter transformation

For the Legendre form of an elliptic curve $y^2=x(x-1)(x-\lambda )$, a change in the $x$ coordinate that permutes its branch values lifts with either of two opposite signs in the $y$ coordinate. These lifts differ by the involution $(x,y)\mapsto (x,-y)$. At $\lambda =e^{\pi i/3}$ the parameter transformation $x\mapsto 1-x/\lambda$ has order three. Its lift fixing $y$ has order three, whereas its lift negating $y$ has order six, since its cube is the nontrivial involution. Thus the order must be computed for a specified lift; both imply the same hexagonal shape of the associated period lattice.