Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - x + 3 x^{2} - 3 x^{3} + 6 x^{4} - 4 x^{5} + 8 x^{6}$ |
| Frobenius angles: | $\pm0.221628809460$, $\pm0.464917083634$, $\pm0.675441718008$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.3307504.1 |
| Galois group: | $S_4\times C_2$ |
| Jacobians: | $2$ |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $10$ | $260$ | $430$ | $5200$ | $44050$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $2$ | $10$ | $8$ | $22$ | $42$ | $70$ | $156$ | $222$ | $386$ | $1050$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):
- $y^2+(x^4+x+1) y=x^8+x^7+x^4+x^3+x^2+1$
- $y^2+(x^4+x+1) y=x^8+x^7+x^3+x^2+x$
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 6.0.3307504.1. |
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_d_d | $2$ | 3.4.f_p_bj |