An **automorphism** of an algebraic curve is an isomorphism from the curve to itself. The set of automorphisms of a curve $X$ form a group $\mathrm{Aut}(X)$ under composition; this is the **automorphism group** of the curve.

The automorphism group of a genus 2 curve necessarily includes the **hyperelliptic involution** $(x,y)\mapsto(x,-y)$, which is an automorphism of order 2; this means that the automorphism group of a genus 2 curve is never trivial.

The geometric automorphism group of a curve $X/k$ is the automorphism group of $X_{\bar k}$.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by John Cremona on 2018-05-23 16:31:29

**Referred to by:**

- ag.cyclic_trigonal
- ag.quotient_curve
- curve.highergenus.aut.full
- curve.highergenus.aut.group_action
- curve.highergenus.aut.groupalgebradecomp
- g2c.geom_aut_grp
- rcs.source.g2c
- lmfdb/genus2_curves/main.py (line 401)
- lmfdb/genus2_curves/main.py (line 636)
- lmfdb/genus2_curves/main.py (line 644)
- lmfdb/genus2_curves/templates/g2c_curve.html (line 123)
- lmfdb/higher_genus_w_automorphisms/templates/hgcwa-index.html (line 24)

**History:**(expand/hide all)

- 2018-05-23 16:31:29 by John Cremona (Reviewed)