Rational Points on a Family of Genus 3 Hyperelliptic Curves

We compute the rational points on certain members of the following family of hyperelliptic curves $$C_a\text{\hspace{0.17em}⁣}:y^2=x^8+(4-4a^4)x^6+(8a^4+6)x^4+(4-4a^4)x^2+1$$ via the method first developed by Dem’yanenko \cite{dem1966rational} and then further generalized by Manin \cite{manin1969p}. In particular, we show that the method of Chabauty–Coleman, while applicable to certain members of this family, is not the most efficient way of computing $C_a(\mathbb{Q})$. We adapt the approach of \cite{kulesz1999application}, incorporating root numbers to further restrict the possible ranks of the elliptic curves arising in the Jacobian decomposition.