Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $6$ |
| L-polynomial: | $( 1 - x + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{2}( 1 - 3 x + 2 x^{2} + x^{3} + 4 x^{4} - 12 x^{5} + 8 x^{6} )$ |
| $1 - 8 x + 31 x^{2} - 75 x^{3} + 127 x^{4} - 170 x^{5} + 224 x^{6} - 340 x^{7} + 508 x^{8} - 600 x^{9} + 496 x^{10} - 256 x^{11} + 64 x^{12}$ | |
| Frobenius angles: | $\pm0.0992589862044$, $\pm0.186455299510$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.757883870938$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $0$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
| $p$-rank: | $4$ |
| Slopes: | $[0, 0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2$ | $5800$ | $712166$ | $81490000$ | $1289229502$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-5$ | $3$ | $16$ | $43$ | $40$ | $78$ | $149$ | $219$ | $511$ | $958$ |
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 $\F_{2^{28}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ac 2 $\times$ 1.2.ab $\times$ 3.2.ad_c_b 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 $\F_{2^{28}}$ is 1.268435456.blhf 4 $\times$ 1.268435456.bwmi 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 $\times$ 3.4.af_s_abp. The endomorphism algebra for each factor is: - 1.4.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 1.4.d : \(\Q(\sqrt{-7}) \).
- 3.4.af_s_abp : \(\Q(\zeta_{7})\).
- Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab $\times$ 1.16.i 2 $\times$ 3.16.l_cg_jf. 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 $\infty$.
- 3.16.l_cg_jf : \(\Q(\zeta_{7})\).
- Endomorphism algebra over $\F_{2^{7}}$
The base change of $A$ to $\F_{2^{7}}$ is 1.128.aq 2 $\times$ 1.128.n 4 . The endomorphism algebra for each factor is: - 1.128.aq 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 1.128.n 4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-7}) \)$)$
- Endomorphism algebra over $\F_{2^{14}}$
The base change of $A$ to $\F_{2^{14}}$ is 1.16384.a 2 $\times$ 1.16384.dj 4 . The endomorphism algebra for each factor is: - 1.16384.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 1.16384.dj 4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-7}) \)$)$
Base change
This is a primitive isogeny class.