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We characterise genus 3 complex smooth hyperelliptic curves that contain twoadditional involutions as curves that can be build from five points in$\mathbb{P}^1$ with a distinguished triple. We are able to write down explicitequations for the curves and all their quotient curves. We show that, fixingone of the elliptic quotient curve, the Prym map becomes a 2:1 map andtherefore the hyperelliptic Klein Prym map, constructed recently by the firstauthor with A. Ortega, is also 2:1 in this case. As a by-product we show anexplicit family of $(1, d)$ polarised abelian surfaces (for d > 1), such thatany surface in the family satisfying a certain explicit condition is abstractlynon-isomorphic to its dual abelian surface.

Hyperelliptic genus 3 curves with involutions and a Prym map