# Automorphisms of curves of arithmetic genus one

Take a projective curve of arithmetic genus one over a finite field $$k=\mathbb{F}_{p^n}$$. The curve is allowed to have nodes but not cusps. How large can the automorphism group of the curve be? For elliptic curves it should be a semidirect product of the group of rational points and the group of automorphism preserving a fixed rational point but I have no mental picture for singular curves.

• If you add rational tails you can make the automorphism as large as you want. – Angelo Aug 9 at 18:42
• @Angelo assume the curve is geometrically irreducible. – hello Aug 9 at 21:20
• If it's geometrically irreducible and has arithmetic genus 1, it is either smooth, or it is a geometrically rational curve with a single node. This last case is not very hard to analyze. – Angelo Aug 13 at 10:15