Nonsingularity away from the singular fibers for Weierstrass inflectionary curves

Let $n=2\ell$, let $F$ have characteristic zero or sufficiently positive characteristic different from three, and let $\mathcal{C}^{\ell}_m\subset\mathbb{P}(1,2,1)$ be the inflectionary curve from $y^n=x^3+\lambda x+2$. Let $\zeta$ be a primitive cube root of unity. Weierstrass nonsingularity conjecture. For every $m\geq3$, $\mathcal{C}^{\ell}_m$ is nonsingular away from the three points $(\zeta^{-j},-3\zeta^j,1)$ for $j\in\{0,1,2\}$. These points correspond to the singular fibers of the Weierstrass pencil; the conjecture predicts that no further singularities occur, and is not resolved in the supplied text.