So, I'm starting out working through Shafarevich's Basic Algebraic Geometry, and while of course it's expected I be familiar with notions from algebra, apparently I have missed a few in my first year courses. In particular, the one that's giving me grief at the moment is that of an 'automorphism of a curve.' In particular, Shafarevich asks me to figure out what the automorphism group of the curve $$y^2 = x^3 + x^2$$
Now I already know in some sense what this should be. The automorphisms should be structure preserving, bijective self-maps of this curve. From clicking around the internet, I gather that the 'structure' here is the property of satisfying $f(x,y) = 0$, i.e. being solutions to the polynomial in two variables given by just shoving everything over to one side. Assuming this is correct, so far so good.
Now the problem is actually computing these automorphisms. My intuition tells me that they should be maps taking $x,y$ to polynomials in $x$ and $y$, and in particular, we want them to have the same image as this equation does. I'm not 100% sure if this is quite the right idea or not, but Shafarevich doesn't seem to define the notion anywhere, so I'm kind of on my own. However, if this fuzzy notion is on the right track, I have been able to spot one such automorphism:
$$ x \mapsto x, \space \space \space y \mapsto -y$$
We can see this is an automorphism, because the negative sign just squares away, and we get back the curve. Now I suspect this is the only automorphism, because intuitively I am thinking of automorphisms of curves as kind of like a group of symmetries of the curve, and we can write $$y^2= x^3 - x^2 = x^2 (x+1)$$
so that this curve has a node at the origin. That makes visualization of this curve not so hard, because between the vertical symmetry found above, and the fact that it has a node, we can kind of see what it looks like. For this reason, I think that the automorphism group, whatever that means, is just $\mathbb Z / 2\mathbb Z$. While aesthetically pleasing, it lacks rigor.
Can anyone help fill me in? It would also be really nice if someone could give me some general guidelines for how to find automorphisms of other curves. Another graduate student has informed me that there are some facts from commutative algebra like 'primary ideal decomposition' and such that are useful. I do not currently know about these things (although I probably should). If your guidelines require any such facts, it would be wonderful if you could please include a reference.
Edit: I forgot to include that I also know that this curve is rational. It can e parameterized by $x = t^2 - 1$, $y= t(t^2-1)$, in case this is useful information.