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

Jyrki Lahtonen
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kre5o
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  • $F_{19}$ or $F_1 9$ – Chinny84 May 25 '15 at 10:19
  • F_{19} of course – kre5o May 25 '15 at 10:23
  • you say of course..but there was an edit that said the latter. So I was making sure. – Chinny84 May 25 '15 at 10:26
  • You should tell us what you have tried. In general we don't like to spoon-feed answers to people wo hasn't shown any effort. – Henrik supports the community May 25 '15 at 10:31
  • It is possible that there is a generic recipe for a curve with $p-1$ points, but I don't remember one (or rather, I remember how to get $p+1$ only). Simple testing shows that $$E:y^2=x^3+x+6$$ works. HOLD ON A SEC. Searching... Nope, wrong recollection. – Jyrki Lahtonen May 25 '15 at 10:37
  • Ok I have tried using PARI/GP and guessing points (x,y) that have square root of y. Tried to somehow connect the fact that its $|E(F_p)|=p+1+\sum\limits{x\in F_p}(f(x)/p)$ – kre5o May 25 '15 at 10:50
  • If you try finding all elliptic curves with nonzero discriminant and check if its order is 18 sooner or later you'll find one that fits, right? – kre5o May 25 '15 at 11:17

1 Answers1

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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.

Jyrki Lahtonen
  • 133,153