Minimal Weierstrass equations for genus 2 curves

We study the minimal Weierstrass equations for genus 2 curves defined over a ring of integers $\mathcal O_{\mathbb F}$. This is done via reduction theory and Julia invariant of binary sextics. We show that when the binary sextics has extra automorphisms this is usually easier to compute. Moreover, we show that when the curve is given in the standard form $y^2=f(x^2)$, where $f(x)$ is a monic polynomial, $f(0)=1$ which is defined over $\mathcal O_{\mathbb F}$ then this form is reduced.