Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + x^{2} + 3 x^{3} + 2 x^{4} + 8 x^{6}$ |
| Frobenius angles: | $\pm0.275395435580$, $\pm0.425423851547$, $\pm0.842895935972$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.4161087.1 |
| Galois group: | $S_4\times C_2$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $15$ | $135$ | $1665$ | $6075$ | $20325$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $7$ | $18$ | $23$ | $18$ | $76$ | $108$ | $263$ | $459$ | $1012$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 0 are hyperelliptic):
- $x^4+x^3 y+x^3 z+x^2 y^2+x z^3+y^3 z=0$
- $x^4+x^3 y+x^3 z+x^2 y^2+x z^3+y^3 z+y^2 z^2=0$
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 6.0.4161087.1. |
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_b_ad | $2$ | 3.4.c_f_l |