Let $C: F(x,y,z)=0$ be the projective curve over $\mathbb{F}_{13}$ given bellow.
$C$ has only two rational points, both singular and the genus of $C$ is one.
To satisfy the Hasse-Weil bound, the desingularization of $C$ must add new points, is this true?
Let $C_2$ be a projective curve over the rationals, is it possible the desingularization to add new points?
Projective Plane Curve over Finite Field of size 13 defined by
x^42 - 3*x^39*z^3 - 4*x^36*y^3*z^3 - x^33*y^6*z^3 - 5*x^30*y^9*z^3 + 2*x^27*y^12*z^3 + x^21*y^18*z^3 - 3*x^18*y^21*z^3 + 2*x^15*y^24*z^3 - 3*x^12*y^27*z^3 + y^39*z^3