Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - x + 2 x^{2} )( 1 - x + x^{2} - 2 x^{3} + 4 x^{4} )$ |
| $1 - 2 x + 4 x^{2} - 5 x^{3} + 8 x^{4} - 8 x^{5} + 8 x^{6}$ | |
| Frobenius angles: | $\pm0.197201053961$, $\pm0.384973271919$, $\pm0.652365995579$ |
| Angle rank: | $3$ (numerical) |
| Jacobians: | $1$ |
| Isomorphism classes: | 5 |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $6$ | $216$ | $504$ | $7344$ | $39666$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $1$ | $9$ | $10$ | $25$ | $41$ | $54$ | $155$ | $305$ | $442$ | $909$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2+(x^4+x^2+x)y=x^8+x^7+x^5+x^4+x$
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_b 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_c_ad | $2$ | 3.4.e_m_x |
| 3.2.a_c_d | $2$ | 3.4.e_m_x |
| 3.2.c_e_f | $2$ | 3.4.e_m_x |