What is the reduction of this hyperelliptic curve

Question

Let $K$ be a number field and $E/K$ an elliptic curve with equation $Y^2Z = X^3 +AXZ^2+BZ^3$ in $\mathbf{P}^2_K$, where $A,B\in K$.

Let $S$ be non-empty finite set of finite places of $K$ and suppose that $E$ has bad reduction over $S$ and good reduction outside $S$. Moreover, let $L/K$ be a finite field extension such that $E_L$ has good reduction over $O_L$. (In particular, $E/K$ has potential good reduction.)

How similar is the reduction of the hyperelliptic curve $H$ of genus $g\geq 2$ given by $$Y^2 Z^{2g-1} = X^{2g+1} + AX Z^{2g} + B Z^{2g+1}$$ to the reduction of $E$?

Does $H_L$ have good reduction over $O_L$?

Does $H$ have good reduction outside $S$?

Does $H$ have bad reduction over $S$?

Answer 1

The first thing you can write already gives a negative answer to all your questions. Take $K=\mathbb{Q}, A=0, B=1, S=\lbrace 2,3 \rbrace$ and $L$ whatever it is the smallest field where $E$ has good reduction. Now take $g=2$. Then $H$ has bad reduction at $5$ and good reduction at $3$, so no to your last two questions. I haven't checked that $H_L$ has bad reduction at $5$ but I see no reason for it to be otherwise.