Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - x + 2 x^{2} )( 1 - x - 2 x^{3} + 4 x^{4} )$ |
| $1 - 2 x + 3 x^{2} - 4 x^{3} + 6 x^{4} - 8 x^{5} + 8 x^{6}$ | |
| Frobenius angles: | $\pm0.139386741866$, $\pm0.384973271919$, $\pm0.686170398078$ |
| Angle rank: | $3$ (numerical) |
| Jacobians: | $2$ |
| Isomorphism classes: | 10 |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4$ | $128$ | $364$ | $6656$ | $30844$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $1$ | $7$ | $7$ | $23$ | $31$ | $55$ | $183$ | $319$ | $511$ | $1047$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which 1 is hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x+1)y=x^7+x^5+x^4+x+1$
- $x^4+x^3y+x^3z+x^2yz+xy^2z+xz^3+y^4=0$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2}$.
Endomorphism algebra over $\F_{2}$| The isogeny class factors as 1.2.ab $\times$ 2.2.ab_a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
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_ae | $2$ | 3.4.c_f_e |
| 3.2.a_b_e | $2$ | 3.4.c_f_e |
| 3.2.c_d_e | $2$ | 3.4.c_f_e |