Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $4$ |
| L-polynomial: | $( 1 - 2 x + 2 x^{2} )( 1 + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )$ |
| $1 - 5 x + 15 x^{2} - 32 x^{3} + 52 x^{4} - 64 x^{5} + 60 x^{6} - 40 x^{7} + 16 x^{8}$ | |
| Frobenius angles: | $\pm0.123548644961$, $\pm0.250000000000$, $\pm0.456881978294$, $\pm0.5$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $0$ |
| Isomorphism classes: | 3 |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3$ | $855$ | $8892$ | $38475$ | $1300233$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-2$ | $10$ | $13$ | $10$ | $38$ | $103$ | $152$ | $210$ | $481$ | $1150$ |
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 $\times$ 1.2.a $\times$ 2.2.ad_f 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 $\times$ 1.16777216.mbf 2 . The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.a $\times$ 1.4.e $\times$ 2.4.b_ad. The endomorphism algebra for each factor is: - 1.4.a : \(\Q(\sqrt{-1}) \).
- 1.4.e : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 2.4.b_ad : \(\Q(\sqrt{-3}, \sqrt{5})\).
- Endomorphism algebra over $\F_{2^{3}}$
The base change of $A$ to $\F_{2^{3}}$ is 1.8.a $\times$ 1.8.e $\times$ 2.8.a_l. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 1.16.i $\times$ 2.16.ah_bh. The endomorphism algebra for each factor is: - 1.16.ai : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 2.16.ah_bh : \(\Q(\sqrt{-3}, \sqrt{5})\).
- Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.a $\times$ 1.64.l 2 $\times$ 1.64.q. The endomorphism algebra for each factor is: - 1.64.a : \(\Q(\sqrt{-1}) \).
- 1.64.l 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
- 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 2 $\times$ 2.256.r_bh. 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 $\infty$.
- 2.256.r_bh : \(\Q(\sqrt{-3}, \sqrt{5})\).
- Endomorphism algebra over $\F_{2^{12}}$
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey $\times$ 1.4096.h 2 $\times$ 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.h 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
- 1.4096.ey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
Base change
This is a primitive isogeny class.