Let $F(x,y,z)$ be the degree 12 homogeneous polynomial:

$$x^{12} - x^9 y^3 + x^6 y^6 - x^3 y^9 + y^{12} - 4 x^9 z^3 + 3 x^6 y^3 z^3 - 2 x^3 y^6 z^3 + y^9 z^3 + 6 x^6 z^6 - 3 x^3 y^3 z^6 + y^6 z^6 - 4 x^3 z^9 + y^3 z^9 + z^{12}$$

Over the rationals it is irreducible and $F=0$ is genus 1 curve.

Numerical evidence in Sagemath and Magma suggests that for infinitely many primes $p$, the curve $F=0$ is irreducible over $\mathbb{F}_p$ and $F=0$ has only one point over $\mathbb{F}_p$, the singular point $(1 : 0 : 1)$.

Q1 Is this true?

Set $p=50033$. Then we have only one point over the finite field and the curve is irreducible of genus 1. This appears to violate the bound on number of rational points over finite fields given in the paper "The number of points on an algebraic curve over a finite field", J.W.P. Hirschfeld, G. Korchmáros and F. Torres ,p. 23.

Q2 What hypothesis am I missing for this violation?

Sagemath code:

```
def tesgfppoints2():
L1=5*10^4
L2=2*L1
for p in primes(L1,L2):
K.<x,y,z>=GF(p)[]
F=x^12 - x^9*y^3 + x^6*y^6 - x^3*y^9 + y^12 - 4*x^9*z^3 + 3*x^6*y^3*z^3 - 2*x^3*y^6*z^3 + y^9*z^3 + 6*x^6*z^6 - 3*x^3*y^3*z^6 + y^6*z^6 - 4*x^3*z^9 + y^3*z^9 + z^12
C=Curve(F)
ire=C.is_irreducible()
if not ire: continue
rp=len(C.rational_points())
print 'p=',p,';rp=',rp,'ir=',ire,'g=',C.genus()
```