Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 2 x )^{2}( 1 + 2 x )^{2}$ |
$1 - 8 x^{2} + 16 x^{4}$ | |
Frobenius angles: | $0$, $0$, $1$, $1$ |
Angle rank: | $0$ (numerical) |
Jacobians: | $0$ |
This isogeny class is not simple, primitive, not ordinary, and supersingular. It is principally polarizable.
Newton polygon
This isogeny class is supersingular.
$p$-rank: | $0$ |
Slopes: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $9$ | $81$ | $3969$ | $50625$ | $1046529$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $1$ | $65$ | $193$ | $1025$ | $3841$ | $16385$ | $64513$ | $262145$ | $1044481$ |
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 $\F_{2^{2}}$The isogeny class factors as 1.4.ae $\times$ 1.4.e 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.ai 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |
Base change
This is a primitive isogeny class.