Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $4$ |
| L-polynomial: | $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 2 x^{2} + 4 x^{4} )$ |
| $1 - 4 x + 6 x^{2} - 8 x^{4} + 24 x^{6} - 32 x^{7} + 16 x^{8}$ | |
| Frobenius angles: | $\pm0.166666666667$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.833333333333$ |
| 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: | $[1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3$ | $225$ | $13689$ | $275625$ | $1669233$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-1$ | $1$ | $17$ | $41$ | $49$ | $97$ | $97$ | $225$ | $449$ | $961$ |
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 $\F_{2}$The isogeny class factors as 1.2.ac 2 $\times$ 2.2.a_ac 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 4 and its endomorphism algebra is $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |
- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.ac 2 $\times$ 1.4.a 2 . The endomorphism algebra for each factor is: - 1.4.ac 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 1.4.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- Endomorphism algebra over $\F_{2^{3}}$
The base change of $A$ to $\F_{2^{3}}$ is 1.8.a 2 $\times$ 1.8.e 2 . The endomorphism algebra for each factor is: - 1.8.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
- 1.8.e 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.e 2 $\times$ 1.16.i 2 . The endomorphism algebra for each factor is: - 1.16.e 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 1.16.i 2 : $\mathrm{M}_{2}(B)$, where $B$ is 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 2 $\times$ 1.64.q 2 . The endomorphism algebra for each factor is: - 1.64.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 1.64.q 2 : $\mathrm{M}_{2}(B)$, where $B$ is 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 2 $\times$ 1.256.q 2 . The endomorphism algebra for each factor is: - 1.256.abg 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 1.256.q 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- Endomorphism algebra over $\F_{2^{12}}$
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey 2 $\times$ 1.4096.ey 2 . The endomorphism algebra for each factor is: - 1.4096.aey 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 1.4096.ey 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
Base change
This is a primitive isogeny class.