Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $6$ |
| L-polynomial: | $( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - 5 x + 12 x^{2} - 20 x^{3} + 29 x^{4} - 40 x^{5} + 48 x^{6} - 40 x^{7} + 16 x^{8} )$ |
| $1 - 8 x + 32 x^{2} - 87 x^{3} + 183 x^{4} - 319 x^{5} + 481 x^{6} - 638 x^{7} + 732 x^{8} - 696 x^{9} + 512 x^{10} - 256 x^{11} + 64 x^{12}$ | |
| Frobenius angles: | $\pm0.0635622003031$, $\pm0.123548644961$, $\pm0.165221137389$, $\pm0.365221137389$, $\pm0.456881978294$, $\pm0.663562200303$ |
| Angle rank: | $3$ (numerical) |
| Jacobians: | $0$ |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $6$ |
| Slopes: | $[0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1$ | $4009$ | $141436$ | $15190101$ | $1005946931$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-5$ | $5$ | $4$ | $13$ | $30$ | $68$ | $198$ | $341$ | $508$ | $1080$ |
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 $\F_{2^{30}}$.
Endomorphism algebra over $\F_{2}$| The isogeny class factors as 2.2.ad_f $\times$ 4.2.af_m_au_bd 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 $\F_{2^{30}}$ is 1.1073741824.acgor 2 $\times$ 2.1073741824.dfu_aggenfjn 2 . The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 2.4.b_ad $\times$ 4.4.ab_c_ai_z. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{2^{3}}$
The base change of $A$ to $\F_{2^{3}}$ is 2.8.a_l $\times$ 4.8.af_d_z_adn. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{2^{5}}$
The base change of $A$ to $\F_{2^{5}}$ is 2.32.ad_bj $\times$ 4.32.a_ad_a_bwv. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.l 2 $\times$ 4.64.at_cz_bur_abapj. The endomorphism algebra for each factor is: - 1.64.l 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
- 4.64.at_cz_bur_abapj : 8.0.13140625.1.
- Endomorphism algebra over $\F_{2^{10}}$
The base change of $A$ to $\F_{2^{10}}$ is 2.1024.ad_bwv 2 $\times$ 2.1024.cj_dzt. The endomorphism algebra for each factor is: - 2.1024.ad_bwv 2 : $\mathrm{M}_{2}($4.0.3625.1$)$
- 2.1024.cj_dzt : \(\Q(\sqrt{-3}, \sqrt{5})\).
- Endomorphism algebra over $\F_{2^{15}}$
The base change of $A$ to $\F_{2^{15}}$ is 2.32768.a_acgor $\times$ 4.32768.a_dfu_a_aggenfjn. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.