Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 2 x + 4 x^{2} )^{2}$ |
| $1 - 4 x + 12 x^{2} - 16 x^{3} + 16 x^{4}$ | |
| Frobenius angles: | $\pm0.333333333333$, $\pm0.333333333333$ |
| Angle rank: | $0$ (numerical) |
| Jacobians: | $1$ |
This isogeny class is not simple, not primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $9$ | $441$ | $6561$ | $74529$ | $986049$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $1$ | $25$ | $97$ | $289$ | $961$ | $3841$ | $16129$ | $66049$ | $264193$ | $1050625$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2+y=x^5+x^4+x^3+a+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{6}}$.
Endomorphism algebra over $\F_{2^{2}}$| The isogeny class factors as 1.4.ac 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
| The base change of $A$ to $\F_{2^{6}}$ is 1.64.q 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 isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.
| Subfield | Primitive Model |
| $\F_{2}$ | 2.2.a_ac |