Invariants
Base field: | $\F_{2}$ |
Dimension: | $5$ |
L-polynomial: | $( 1 - 2 x + 2 x^{2} )( 1 - 4 x + 5 x^{2} + 2 x^{3} - 11 x^{4} + 4 x^{5} + 20 x^{6} - 32 x^{7} + 16 x^{8} )$ |
$1 - 6 x + 15 x^{2} - 16 x^{3} - 5 x^{4} + 30 x^{5} - 10 x^{6} - 64 x^{7} + 120 x^{8} - 96 x^{9} + 32 x^{10}$ | |
Frobenius angles: | $\pm0.0247483856139$, $\pm0.177336015878$, $\pm0.250000000000$, $\pm0.344002682545$, $\pm0.858081718947$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $0$ |
Isomorphism classes: | 1 |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $4$ |
Slopes: | $[0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1$ | $305$ | $65572$ | $1923025$ | $27087101$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-3$ | $-1$ | $15$ | $27$ | $27$ | $65$ | $81$ | $259$ | $447$ | $959$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{12}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ac $\times$ 4.2.ae_f_c_al and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey $\times$ 2.4096.ahm_zkj 2 . The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.a $\times$ 4.4.ag_t_abq_dd. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{2^{3}}$
The base change of $A$ to $\F_{2^{3}}$ is 1.8.e $\times$ 4.8.c_c_ak_aep. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.i $\times$ 4.16.c_t_adq_adr. The endomorphism algebra for each factor is: - 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 4.16.c_t_adq_adr : 8.0.22581504.2.
- Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.a $\times$ 4.64.a_ahm_a_zkj. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.