Invariants
Base field: | $\F_{2}$ |
Dimension: | $6$ |
L-polynomial: | $( 1 - 2 x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - 3 x + 2 x^{2} + x^{3} + 4 x^{4} - 12 x^{5} + 8 x^{6} )$ |
$1 - 8 x + 30 x^{2} - 70 x^{3} + 117 x^{4} - 161 x^{5} + 218 x^{6} - 322 x^{7} + 468 x^{8} - 560 x^{9} + 480 x^{10} - 256 x^{11} + 64 x^{12}$ | |
Frobenius angles: | $\pm0.0992589862044$, $\pm0.123548644961$, $\pm0.186455299510$, $\pm0.250000000000$, $\pm0.456881978294$, $\pm0.757883870938$ |
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: | $5$ |
Slopes: | $[0, 0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1$ | $2755$ | $297388$ | $34836975$ | $1373558261$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-5$ | $1$ | $7$ | $29$ | $40$ | $109$ | $191$ | $237$ | $538$ | $1076$ |
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 $\F_{2^{84}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ac $\times$ 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 $\F_{2^{84}}$ is 1.19342813113834066795298816.auojdvfkpl 3 $\times$ 1.19342813113834066795298816.aptgzwqopl 2 $\times$ 1.19342813113834066795298816.bqdecdesiy. 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 $\times$ 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 1.8.e $\times$ 2.8.a_l $\times$ 3.8.ag_bd_adf. 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.i $\times$ 2.16.ah_bh $\times$ 3.16.l_cg_jf. The endomorphism algebra for each factor is: - 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 2.16.ah_bh : \(\Q(\sqrt{-3}, \sqrt{5})\).
- 3.16.l_cg_jf : \(\Q(\zeta_{7})\).
- Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.a $\times$ 1.64.l 2 $\times$ 3.64.w_lx_ekt. The endomorphism algebra for each factor is: - 1.64.a : \(\Q(\sqrt{-1}) \).
- 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.aq $\times$ 1.128.n 3 $\times$ 2.128.bn_yl. The endomorphism algebra for each factor is: - 1.128.aq : \(\Q(\sqrt{-1}) \).
- 1.128.n 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
- 2.128.bn_yl : \(\Q(\sqrt{-3}, \sqrt{5})\).
- Endomorphism algebra over $\F_{2^{12}}$
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.h 2 $\times$ 1.4096.ey $\times$ 3.4096.fe_fpl_agrpx. The endomorphism algebra for each factor is: - 1.4096.h 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
- 1.4096.ey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 3.4096.fe_fpl_agrpx : \(\Q(\zeta_{7})\).
- Endomorphism algebra over $\F_{2^{14}}$
The base change of $A$ to $\F_{2^{14}}$ is 1.16384.a $\times$ 1.16384.dj 3 $\times$ 2.16384.ajr_cqyz. The endomorphism algebra for each factor is: - 1.16384.a : \(\Q(\sqrt{-1}) \).
- 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 $\times$ 1.2097152.dau $\times$ 2.2097152.a_hpued. The endomorphism algebra for each factor is: - 1.2097152.aedn 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
- 1.2097152.dau : \(\Q(\sqrt{-1}) \).
- 2.2097152.a_hpued : \(\Q(\sqrt{-3}, \sqrt{5})\).
- Endomorphism algebra over $\F_{2^{28}}$
The base change of $A$ to $\F_{2^{28}}$ is 1.268435456.blhf 3 $\times$ 1.268435456.bwmi $\times$ 2.268435456.bssv_ccitvwj. The endomorphism algebra for each factor is: - 1.268435456.blhf 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
- 1.268435456.bwmi : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 2.268435456.bssv_ccitvwj : \(\Q(\sqrt{-3}, \sqrt{5})\).
- Endomorphism algebra over $\F_{2^{42}}$
The base change of $A$ to $\F_{2^{42}}$ is 1.4398046511104.ahxvrd 3 $\times$ 1.4398046511104.a $\times$ 1.4398046511104.hpued 2 . The endomorphism algebra for each factor is: - 1.4398046511104.ahxvrd 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
- 1.4398046511104.a : \(\Q(\sqrt{-1}) \).
- 1.4398046511104.hpued 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
Base change
This is a primitive isogeny class.