Invariants
Base field: | $\F_{2}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 2 x + 2 x^{2} )^{2}( 1 - x + x^{2} - 2 x^{3} + 4 x^{4} )$ |
$1 - 5 x + 13 x^{2} - 22 x^{3} + 32 x^{4} - 44 x^{5} + 52 x^{6} - 40 x^{7} + 16 x^{8}$ | |
Frobenius angles: | $\pm0.197201053961$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.652365995579$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $0$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3$ | $675$ | $6084$ | $286875$ | $3030843$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-2$ | $6$ | $13$ | $42$ | $68$ | $63$ | $110$ | $210$ | $373$ | $966$ |
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 2 $\times$ 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:
|
The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 2 $\times$ 2.16.j_bx. 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 2 $\times$ 2.4.b_f. The endomorphism algebra for each factor is: - 1.4.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 2.4.b_f : 4.0.2873.1.
Base change
This is a primitive isogeny class.