Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + x + x^{2} + 3 x^{3} + 2 x^{4} + 4 x^{5} + 8 x^{6}$ |
| Frobenius angles: | $\pm0.265053999545$, $\pm0.521842670017$, $\pm0.907278725697$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.503792.1 |
| Galois group: | $S_4\times C_2$ |
| Jacobians: | $3$ |
| Isomorphism classes: | 5 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $20$ | $80$ | $1340$ | $3200$ | $49100$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $6$ | $16$ | $14$ | $44$ | $78$ | $88$ | $222$ | $484$ | $1166$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which 2 are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^4+x+1)y=x^5+x^2+x$
- $y^2+(x^4+x)y=x^7+x^5+x^2+x$
- $x^3z+x^2y^2+xyz^2+xz^3+y^3z=0$
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 6.0.503792.1. |
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.ab_b_ad | $2$ | 3.4.b_ab_d |