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

Question

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 ?

Answer 1

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.