How to calculate an elliptic curve

Question

I need to find an elliptic curve in $F_{19}$ that has $|E(F_{19})|=18$. I am really stuck here. Can anyone help?

Answer 1

Searching for curves of the form $y^2=x^3+x+k$ gave me the following hit. With $k=6$, the value of $f(x)=x^3+x+6$ is zero for $x=6$ (1 point), and $f(x)$ is a non-zero square of $\Bbb{F}_{19}$ for eight choices $x=0,2,3,4,10,12,14,18$ (16 points). Including the point at infinity gives us a total of 18 points.

---

The Mathematica snippet

counter = 1; For[y = 0, y < 19, ++y, t = Mod[y^2, 19];
 For[x = 0, x < 19, ++x, 
  If[Mod[x^3 + x + 6, 19] == t, ++counter]]]; counter

was used. No need to do anything fancier than a brute force double loop with such a small field.