Invariants
Base field: | $\F_{2}$ |
Dimension: | $5$ |
L-polynomial: | $( 1 - 2 x + 2 x^{2} )( 1 - 4 x + 6 x^{2} - 4 x^{3} + 2 x^{4} - 8 x^{5} + 24 x^{6} - 32 x^{7} + 16 x^{8} )$ |
$1 - 6 x + 16 x^{2} - 24 x^{3} + 22 x^{4} - 20 x^{5} + 44 x^{6} - 96 x^{7} + 128 x^{8} - 96 x^{9} + 32 x^{10}$ | |
Frobenius angles: | $\pm0.0377785699724$, $\pm0.148391828106$, $\pm0.250000000000$, $\pm0.398391828106$, $\pm0.787778569972$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $0$ |
Isomorphism classes: | 7 |
This isogeny class is not simple, primitive, not ordinary, and not supersingular.
Newton polygon
$p$-rank: | $0$ |
Slopes: | $[1/4, 1/4, 1/4, 1/4, 1/2, 1/2, 3/4, 3/4, 3/4, 3/4]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1$ | $485$ | $34957$ | $1690225$ | $18761641$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-3$ | $1$ | $9$ | $25$ | $17$ | $73$ | $137$ | $241$ | $513$ | $881$ |
Jacobians and polarizations
This isogeny class does not contain a Jacobian, and it is unknown whether it is principally polarizable.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{8}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ac $\times$ 4.2.ae_g_ae_c 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 $\F_{2^{8}}$ is 1.256.abg $\times$ 4.256.q_ds_glk_jitk. The endomorphism algebra for each factor is:
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- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.a $\times$ 4.4.ae_i_ai_e. 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.a_i_a_q. The endomorphism algebra for each factor is: - 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 4.16.a_i_a_q : 8.0.18939904.2.
Base change
This is a primitive isogeny class.