Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - x - x^{2} + 2 x^{3} - 2 x^{4} - 4 x^{5} + 8 x^{6}$ |
| Frobenius angles: | $\pm0.0743927619408$, $\pm0.403891304032$, $\pm0.869099747587$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.6041764.1 |
| Galois group: | $S_4\times C_2$ |
| Jacobians: | $1$ |
| Isomorphism classes: | 2 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3$ | $27$ | $684$ | $3051$ | $17193$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $2$ | $2$ | $11$ | $10$ | $12$ | $71$ | $128$ | $322$ | $479$ | $1042$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is not hyperelliptic), and hence is principally polarizable:
- $x^3y+xy^3+xy^2z+xz^3+y^4=0$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2}$.
Endomorphism algebra over $\F_{2}$| The endomorphism algebra of this simple isogeny class is 6.0.6041764.1. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 3.2.b_ab_ac | $2$ | 3.4.ad_b_i |