Invariants
Base field: | $\F_{2}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - x + 2 x^{2} )( 1 - 2 x + 3 x^{2} - 4 x^{3} + 4 x^{4} )$ |
$1 - 3 x + 7 x^{2} - 11 x^{3} + 14 x^{4} - 12 x^{5} + 8 x^{6}$ | |
Frobenius angles: | $\pm0.174442860055$, $\pm0.384973271919$, $\pm0.546783656212$ |
Angle rank: | $3$ (numerical) |
Jacobians: | $1$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $4$ | $224$ | $868$ | $3584$ | $38764$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $0$ | $10$ | $12$ | $14$ | $40$ | $82$ | $140$ | $286$ | $552$ | $930$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2+(x^4+x+1)y=x^8+x^6+x^5+x^3+x^2+x+1$
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.ac_d 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.ab_d_af | $2$ | 3.4.f_l_t |
3.2.b_d_f | $2$ | 3.4.f_l_t |
3.2.d_h_l | $2$ | 3.4.f_l_t |