Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 2 x^{2} + x^{3} + 4 x^{4} + 8 x^{6}$ |
| Frobenius angles: | $\pm0.271447383401$, $\pm0.471364905773$, $\pm0.768857555471$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.6660007.1 |
| Galois group: | $S_4\times C_2$ |
| Jacobians: | $3$ |
| Isomorphism classes: | 4 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $16$ | $224$ | $832$ | $5824$ | $24496$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $9$ | $12$ | $25$ | $23$ | $78$ | $129$ | $177$ | $516$ | $1069$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which 2 are hyperelliptic):
- $y^2+(x^4+x^2+x+1) y=x^7+1$
- $y^2+(x^4+x^2+x+1) y=x^8+x^6+x^5+x^4$
- $x^4+x^3 z+x^2 y z+x y^3+x y^2 z+x z^3+y^2 z^2=0$
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 6.0.6660007.1. |
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.a_c_ab | $2$ | 3.4.e_m_bf |