Why does MAGMA claim that the automorphism group of an elliptic curve is order 24 when it is order 12?

Question

I am trying to get the hang of the available software for computing automorphism groups of plane curves over finite fields. I am using this Magma code to test it out on $y^2 = x^3 - x$ over $\mathbb{F}_3$, which we know is of order 12 by (pg. 410, Prop 1.2(c)) Silverman's Arithmetic of Elliptic Curves.

The following Magma code gives me as output that the order is 24. Moreover, it also does this for seemingly every other plane curve I can think to put in.

I wrote this code according to the exposition here on the Magma webpage. Something terribly wrong is going on, likely I am misunderstanding something fundamental.

A<x,y> := AffineSpace(FiniteField(3),2);
f := y^2 - x^3 + x;
C := Curve(A,f);
G := AutomorphismGroup(C);
Order(G);

Answer 1

Magma is computing the automorphism group of the associated projective curve $E$ defined over the base field (according to the link you gave). You are thinking of the automorphism group of $E$ as an elliptic curve i.e. with $\infty$ fixed. The group magma is computing will also include the translations which are defined over the base, so there is a short exact sequence:

$$0 \rightarrow E(k) \rightarrow \mathrm{AutAsProjectiveCurve}(E,k) \rightarrow \mathrm{AutAsEllipticCurve}(E,k) \rightarrow 0.$$

Your curve has $4$ points defined over $\mathbf{F}_3$, but in fact only a subgroup of order $6$ of the order $12$ group $\mathrm{AutAsEllipticCurve}(E,\overline{\mathbf{F}}_3)$ of geometric automorphisms fixing $\infty$ are defined over $\mathbf{F}_3$. Thus $24 = 6 \cdot 4$ is the correct anser. The extra automorphism of the elliptic curve is defined over $\mathbf{F}_9 = \mathbf{F}_3[i]$. Also, we have $|E(\mathbf{F}_9)| = 16$, because the the zeta function is equal to

$$\frac{(1 + 3 T^2)}{(1-T)(1-3T)} = 1 + 4 T + 16 T^2 + \ldots = 1 + 4 T + \frac{1}{2} \left(\frac{4^2}{2} + 16\right) T^2 + \ldots $$

And indeed:

A<x,y> := AffineSpace(FiniteField(9),2);
f := y^2 - x^3 + x;
C := Curve(A,f);
G := AutomorphismGroup(C);
Order(G);

Gives the answer $192 = 12 \cdot 16$. This is also why magma will refuse to do the same computation over $\mathbf{Q}$ --- that would require computing all the rational points! To give some other examples, you can take a random elliptic curve with no automorphisms (as an elliptic curve) besides $\pm 1$ and compare the answer to the number of points modulo p:

A<x,y> := AffineSpace(FiniteField(41),2);
f := y^2 - (x^3 + x + 1);
C := Curve(A,f);
G := AutomorphismGroup(C);
Order(G);

returns $70$, and indeed $|A(\mathbf{F}_{41})| = 1 + 41 - (-7) = 35$, and $70 = 2 \cdot 35$.