I am studying the following quartic curve:
$f(x,y) = c_1x^2 + c_2x^4 + c_3x^2y + c_4x^2y^2 + c_5y^2 + c_ 6y^3 + c_7y^4$
where $c_i$ are constant (in fact they are expressions in terms of other constants). Starting to learn a bit about curves, I found that a necessary condition for a point $(x_0, y_0)$ be singular (a double point) is that
$$F(x_0, y_0) = 0,\qquad F_x (x_0, y_0) = 0,\qquad F_y (x_0, y_0) = 0$$
and that the second derivatives calculated at that point are not all equal to zero. Solving these three equations (trial and error) I got two solutions:
$$(x_0, y_0) = (0,0),\qquad (x_0, y_0) = (0, -2 c_5/c_6)$$
The second solution is a solution due to the fact that the coefficients $c_i$ are interrelated. For both points the second derivatives are not equal to zero.
Therefore, this curve has apparently has two double points, both with multiplicity equal to 2. Thus, this curve would have genus = 1, if there are no more singular points.
My questions are:
1) What I said above accurate?
2) is there any simple way to test if there is more singular points?
3) if no more singular points, how to parameterize a quartic curve like that? (I tried to transform this curve in an elliptic one, making $x^2 = z$, but i'm not sure if this is correct.)
Thank you very much