Geometric irreducibility and genus conjecture for Weierstrass inflectionary curves

Let $n=2\ell$, let $F$ have characteristic zero or sufficiently positive characteristic, and let $\mathcal{C}^{\ell}_m\subset\mathbb{P}(1,2,1)$ be the inflectionary curve derived from $y^n=x^3+\lambda x+2$. Weierstrass genus conjecture. For every $m\geq3$, $\mathcal{C}^{\ell}_m$ is geometrically irreducible and has geometric genus $\left\lceil\frac{(m-1)^2}{4}\right\rceil$. The claim is intended to follow from the predicted singularity structure and Newton polygons; the supplied text does not establish it.