Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 3 x + 3 x^{2} + 2 x^{3} + 12 x^{4} - 48 x^{5} + 64 x^{6}$ |
| Frobenius angles: | $\pm0.177291589221$, $\pm0.277218038530$, $\pm0.766593115976$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 6.0.2101707.2 |
| Galois group: | $D_{6}$ |
| Jacobians: | $4$ |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $31$ | $3999$ | $291772$ | $22894275$ | $1101883561$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $2$ | $14$ | $71$ | $338$ | $1052$ | $4199$ | $16424$ | $64802$ | $262799$ | $1047554$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which 0 are hyperelliptic):
- $x^4+a x^3 z+a x^2 y^2+x z^3+a y^4+y^3 z+y^2 z^2=0$
- $a^2 x^4+x^3 z+a^2 x^2 y^2+x^2 y z+x z^3+a y^4+y^3 z+y^2 z^2=0$
- $a x^4+a x^3 z+a^2 x^2 y^2+a^2 x^2 z^2+x z^3+a y^4+y^3 z+y^2 z^2=0$
- $a^2 x^4+a^2 x^3 y+a x^3 z+x^2 y^2+a x^2 y z+x^2 z^2+a x y z^2+x z^3+a y^4+y^3 z+y^2 z^2=0$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{2}}$.
Endomorphism algebra over $\F_{2^{2}}$| The endomorphism algebra of this simple isogeny class is 6.0.2101707.2. |
Base change
This is a primitive isogeny class.