Exact output statistics of Icart's encoding in the exceptional $j=0$ case

Icart’s encoding is a classical deterministic map from finite fields to elliptic curves and a basic ingredient in early hash-to-curve constructions. We determine the exact one-output distribution of this map in the exceptional $j=0$ case. More precisely, for $$E_{0,b}:Y^2=X^3+b,q\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}3),$$ we compute the complete fibre distribution of $f_{0,b}:{\mathbb{F}}_q\to E_{0,b}({\mathbb{F}}_q)$. This gives closed formulae for the image size, total variation distance from uniform, collision probability, power sums, entropy measures and basic batch statistics. We also derive the exact second moment of all nontrivial character sums of the output distribution. Via the Weil pairing, this becomes an exact energy formula for pairing-character tests on the supersingular $j=0$ family whose odd prime order subgroups have embedding degree two.