Jacobian of the Picard Curve

Jack Quine

We compute the Jacobian of the genus $3$ curve with $48$ automorphisms and show that the minimum non-zero norm is $1 + \sqrt3/3$. This show that Klein's surface with with minimum non-zero norm $4/\sqrt7$ is not extremal for genus 3. We show that the lattice obtained is the isodual lattice $M(E_6)$ discovered by Conway and Sloane. Our computation gives a cyclotomic construction of this lattice.