Invariants
Base field: | $\F_{2}$ |
Dimension: | $5$ |
L-polynomial: | $( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - 3 x + 2 x^{2} + x^{3} + 4 x^{4} - 12 x^{5} + 8 x^{6} )$ |
$1 - 6 x + 16 x^{2} - 26 x^{3} + 33 x^{4} - 43 x^{5} + 66 x^{6} - 104 x^{7} + 128 x^{8} - 96 x^{9} + 32 x^{10}$ | |
Frobenius angles: | $\pm0.0992589862044$, $\pm0.123548644961$, $\pm0.186455299510$, $\pm0.456881978294$, $\pm0.757883870938$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $0$ |
Isomorphism classes: | 1 |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $5$ |
Slopes: | $[0, 0, 0, 0, 0, 1, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1$ | $551$ | $22876$ | $1393479$ | $33501421$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-3$ | $1$ | $3$ | $21$ | $32$ | $109$ | $207$ | $269$ | $570$ | $1076$ |
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 $\F_{2^{42}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 2.2.ad_f $\times$ 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 $\F_{2^{42}}$ is 1.4398046511104.ahxvrd 3 $\times$ 1.4398046511104.hpued 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 $\times$ 3.4.af_s_abp. 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 $\times$ 3.8.ag_bd_adf. 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 $\times$ 3.64.w_lx_ekt. The endomorphism algebra for each factor is: - 1.64.l 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
- 3.64.w_lx_ekt : \(\Q(\zeta_{7})\).
- Endomorphism algebra over $\F_{2^{7}}$
The base change of $A$ to $\F_{2^{7}}$ is 1.128.n 3 $\times$ 2.128.bn_yl. The endomorphism algebra for each factor is: - 1.128.n 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
- 2.128.bn_yl : \(\Q(\sqrt{-3}, \sqrt{5})\).
- Endomorphism algebra over $\F_{2^{14}}$
The base change of $A$ to $\F_{2^{14}}$ is 1.16384.dj 3 $\times$ 2.16384.ajr_cqyz. The endomorphism algebra for each factor is: - 1.16384.dj 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
- 2.16384.ajr_cqyz : \(\Q(\sqrt{-3}, \sqrt{5})\).
- Endomorphism algebra over $\F_{2^{21}}$
The base change of $A$ to $\F_{2^{21}}$ is 1.2097152.aedn 3 $\times$ 2.2097152.a_hpued. The endomorphism algebra for each factor is: - 1.2097152.aedn 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
- 2.2097152.a_hpued : \(\Q(\sqrt{-3}, \sqrt{5})\).
Base change
This is a primitive isogeny class.