Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 2 x + 2 x^{2} - 2 x^{3} + 4 x^{4} - 8 x^{5} + 8 x^{6}$ |
| Frobenius angles: | $\pm0.111901318694$, $\pm0.359194778829$, $\pm0.729359314356$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.1142512.1 |
| Galois group: | $S_4\times C_2$ |
| Jacobians: | $2$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3$ | $81$ | $387$ | $7209$ | $23493$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $1$ | $5$ | $7$ | $25$ | $21$ | $53$ | $169$ | $289$ | $601$ | $1105$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which 1 is hyperelliptic), and hence is principally polarizable:
- $y^2+y=x^7+x^6+1$
- $x^4+x^2y^2+x^2z^2+xz^3+y^4+y^3z+y^2z^2=0$
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 6.0.1142512.1. |
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_c_c | $2$ | 3.4.a_e_ae |