Invariants
Base field: | $\F_{2}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 2 x + 2 x^{2} )( 1 - x + 3 x^{2} - 2 x^{3} + 4 x^{4} )$ |
$1 - 3 x + 7 x^{2} - 10 x^{3} + 14 x^{4} - 12 x^{5} + 8 x^{6}$ | |
Frobenius angles: | $\pm0.250000000000$, $\pm0.306143893905$, $\pm0.570118980449$ |
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: | $2$ |
Slopes: | $[0, 0, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $5$ | $275$ | $1040$ | $6875$ | $56375$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $0$ | $10$ | $15$ | $26$ | $50$ | $55$ | $70$ | $226$ | $555$ | $1050$ |
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 $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ac $\times$ 2.2.ab_d 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 $\F_{2^{4}}$ is 1.16.i $\times$ 2.16.b_b. 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 $\times$ 2.4.f_n. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.