Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 2 x + 4 x^{2} - 6 x^{3} + 8 x^{4} - 8 x^{5} + 8 x^{6}$ |
| Frobenius angles: | $\pm0.159385776661$, $\pm0.421684666515$, $\pm0.635759575232$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.503792.1 |
| 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: | $0$ |
| Slopes: | $[1/3, 1/3, 1/3, 2/3, 2/3, 2/3]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $5$ | $185$ | $395$ | $4625$ | $40025$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $1$ | $9$ | $7$ | $17$ | $41$ | $69$ | $169$ | $321$ | $457$ | $929$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2+y=x^7+x^4+1$
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 6.0.503792.1. |
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.c_e_g | $2$ | 3.4.e_i_m |