0

Possible cardinalities of an elliptic curve over $\mathbb F_7$

Where the equation is given by $y^2=x^3+ax+b$. Hasse Bound gives something between $3$ and $13$ but my question is, can all these values be attained ? Playing around here a bit, I got for example,

$y^2=x^3+1$ ($12$ points)

$y^2=x^3+2$ ($9$ points)

$y^2=x^3+3$ ($13$ points)

$y^2=x^3+4$ ($3$ points)

but what can I say for the rest ?

user257
  • 989

1 Answers1

0

Brute force:

The entry in the following table at row $a$ and column $b$, each from $0$ to $6$, is the number of points in the affine part of the curve $y^2=x^3+ax+b$.

7   11   8   12   2   6   3
7   4    8   5    9   6   10
7   4    8   5    9   6   10
7   11   8   5    9   6   3
7   4    8   5    9   6   10
7   11   8   5    9   6   3
7   11   8   5    9   6   3

I computed this with this Mathematica code:

eq[a_, b_][{x_, y_}] := Mod[y^2 - x^3 - a x - b, 7] == 0
count[a_, b_] :=  Length@ Select[Tuples[Range[0, 6], 2], eq[a, b]]
Table[count[a, b], {a, 0, 6}, {b, 0, 6}]

Notice that some of the entries do not correspond to elliptic curves.

Mariano Suárez-Álvarez
  • 135,076
  • Thanks but I think you forgot to count the point at infinite somwhere, e.g. for $y^2=x^3+1; (0,1) (0,6) (\infty) (1,3) (3,0) (1,4) (6,0) (2,4) (2,3) (5,0) (4,4) (4,3)$ I got ($12$ points) and for the $0$th row $1$st column $9$ instead of $8$ – user257 Nov 23 '16 at 22:17
  • 1
    That is precisely what I mean by «the points in the affine part of the curve». – Mariano Suárez-Álvarez Nov 23 '16 at 22:18