Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 3 x + 5 x^{2} + 7 x^{3} + 10 x^{4} + 12 x^{5} + 8 x^{6}$ |
| Frobenius angles: | $\pm0.358750840369$, $\pm0.683161207432$, $\pm0.894721499061$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.679024.1 |
| Galois group: | $S_4\times C_2$ |
| Jacobians: | $1$ |
| Isomorphism classes: | 1 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $46$ | $92$ | $598$ | $6256$ | $27646$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $6$ | $12$ | $22$ | $26$ | $42$ | $132$ | $318$ | $462$ | $1126$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2+(x^4+x+1)y=x^7+x^5+x^4+x^3+x^2+x$
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 6.0.679024.1. |
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.ad_f_ah | $2$ | 3.4.b_d_af |