  In this paper we give a passage formula between different invariants of genus
3 hyperelliptic curves: in particular between Tsuyumine and Shioda invariants.
This is needed to get modular expressions for Shioda invariants, that is, for
example, useful for proving the correctness of numerically computed equations
of CM genus 3 hyperelliptic curves.
  On the other hand, we also get Shioda invariants described in terms of
differences of roots of the equation defining the hyperelliptic curve, that has
applications for studying the reduction type of the curve. Under certain
conditions on its jacobian, we give a criterion for determining the type of bad
reduction of a genus 3 hyperelliptic curve.
