Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $5$ |
| L-polynomial: | $( 1 - x + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{2}( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )$ |
| $1 - 8 x + 34 x^{2} - 97 x^{3} + 204 x^{4} - 328 x^{5} + 408 x^{6} - 388 x^{7} + 272 x^{8} - 128 x^{9} + 32 x^{10}$ | |
| Frobenius angles: | $\pm0.123548644961$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.456881978294$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $0$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
| $p$-rank: | $3$ |
| Slopes: | $[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2$ | $3800$ | $179816$ | $1710000$ | $35539702$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-5$ | $9$ | $22$ | $25$ | $35$ | $78$ | $149$ | $241$ | $454$ | $1029$ |
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 $\F_{2^{12}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ac 2 $\times$ 1.2.ab $\times$ 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 $\F_{2^{12}}$ is 1.4096.h 2 $\times$ 1.4096.bv $\times$ 1.4096.ey 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 2 $\times$ 1.4.d $\times$ 2.4.b_ad. The endomorphism algebra for each factor is: - 1.4.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 1.4.d : \(\Q(\sqrt{-7}) \).
- 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.e 2 $\times$ 1.8.f $\times$ 2.8.a_l. The endomorphism algebra for each factor is: - 1.8.e 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 1.8.f : \(\Q(\sqrt{-7}) \).
- 2.8.a_l : \(\Q(\sqrt{-3}, \sqrt{5})\).
- Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab $\times$ 1.16.i 2 $\times$ 2.16.ah_bh. The endomorphism algebra for each factor is: - 1.16.ab : \(\Q(\sqrt{-7}) \).
- 1.16.i 2 : $\mathrm{M}_{2}(B)$, where $B$ is 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.aj $\times$ 1.64.a 2 $\times$ 1.64.l 2 . The endomorphism algebra for each factor is: - 1.64.aj : \(\Q(\sqrt{-7}) \).
- 1.64.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 1.64.l 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
Base change
This is a primitive isogeny class.