A function field over F_q is isomorphic to L(Λ_{x²}) exactly when it has a subgroup isomorphic to (F_q,+) × F_q^*, genus 1 + q(q-3)/2, and exactly q+1 rational places.
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A Note on Cyclotomic Function Fields with Quadratic Modulus
A function field over F_q is isomorphic to L(Λ_{x²}) exactly when it has a subgroup isomorphic to (F_q,+) × F_q^*, genus 1 + q(q-3)/2, and exactly q+1 rational places.