Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 2 x + 4 x^{2} )( 1 + 4 x^{2} )$ |
| $1 - 2 x + 8 x^{2} - 8 x^{3} + 16 x^{4}$ | |
| Frobenius angles: | $\pm0.333333333333$, $\pm0.5$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $15$ | $525$ | $5265$ | $61425$ | $1017825$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $29$ | $81$ | $241$ | $993$ | $4097$ | $16257$ | $65281$ | $263169$ | $1051649$ |
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 $\F_{2^{24}}$.
Endomorphism algebra over $\F_{2^{2}}$| The isogeny class factors as 1.4.ac $\times$ 1.4.a 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 $\F_{2^{24}}$ is 1.16777216.amdc 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |
- Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.e $\times$ 1.16.i. The endomorphism algebra for each factor is: - 1.16.e : \(\Q(\sqrt{-3}) \).
- 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.a $\times$ 1.64.q. The endomorphism algebra for each factor is: - 1.64.a : \(\Q(\sqrt{-1}) \).
- 1.64.q : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- Endomorphism algebra over $\F_{2^{8}}$
The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg $\times$ 1.256.q. The endomorphism algebra for each factor is: - 1.256.abg : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 1.256.q : \(\Q(\sqrt{-3}) \).
- Endomorphism algebra over $\F_{2^{12}}$
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey $\times$ 1.4096.ey. The endomorphism algebra for each factor is: - 1.4096.aey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 1.4096.ey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
Base change
This is a primitive isogeny class.