  We develop the formalism of the first order near-axis expansion of the MHD
equilibrium equations described in Garren & Boozer (1991), Plunk et al. (2019)
and Plunk et al. (2021), for the case of a quasi-isodynamic, N-field period,
stellarator symmetric, single-well magnetic field equilibrium. The importance
of the magnetic axis shape is investigated, and we conclude that control of the
curvature and torsion is crucial to obtain omnigenous configurations with
finite aspect ratio and low effective ripple, especially for a higher number of
field periods. For this reason a method is derived to construct classes of axis
shapes with favourable curvature and torsion. Solutions are presented,
including a three-field-period configuration constructed at an aspect ratio of
A=20, with a maximum elongation of e=3.2 and an effective ripple under 1%,
which demonstrates that high elongation is not a necessary feature of QI
stellarators.
