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 : y 2 = x 8 + ( 4 − 4 a 4 ) x 6 + ( 8 a 4 + 6 ) x 4 + ( 4 − 4 a 4 ) x 2 + 1 via the method first developed by Dem'yanenko [Dem68] and then further generalized by Manin [Man69] . 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 ( Q ) . We adapt the approach of [Kul99] , incorporating root numbers to further restrict the possible ranks of the elliptic curves arising in the Jacobian decomposition.