Invariants
| Base field: | $\F_{2^{6}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 8 x )^{2}( 1 - 8 x + 64 x^{2} )$ |
| $1 - 24 x + 256 x^{2} - 1536 x^{3} + 4096 x^{4}$ | |
| Frobenius angles: | $0$, $0$, $\pm0.333333333333$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2793$ | $16515009$ | $68718952449$ | $281406257229825$ | $1152815955785449473$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $41$ | $4033$ | $262145$ | $16773121$ | $1073643521$ | $68718428161$ | $4398040219649$ | $281474959933441$ | $18014398509481985$ | $1152921503533105153$ |
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 $\F_{2^{36}}$.
Endomorphism algebra over $\F_{2^{6}}$The isogeny class factors as 1.64.aq $\times$ 1.64.ai 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 $\F_{2^{36}}$ is 1.68719476736.abdvoy 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 $\F_{2^{12}}$
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey $\times$ 1.4096.cm. The endomorphism algebra for each factor is: - 1.4096.aey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 1.4096.cm : \(\Q(\sqrt{-3}) \).
- Endomorphism algebra over $\F_{2^{18}}$
The base change of $A$ to $\F_{2^{18}}$ is 1.262144.abnk $\times$ 1.262144.bnk. The endomorphism algebra for each factor is: - 1.262144.abnk : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 1.262144.bnk : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
Base change
This is a primitive isogeny class.