Invariants
Base field: | $\F_{2}$ |
Dimension: | $6$ |
L-polynomial: | $( 1 - 2 x + 2 x^{2} )( 1 - 2 x + 2 x^{2} - 4 x^{3} + 4 x^{4} )( 1 - 4 x + 9 x^{2} - 15 x^{3} + 18 x^{4} - 16 x^{5} + 8 x^{6} )$ |
$1 - 8 x + 33 x^{2} - 95 x^{3} + 214 x^{4} - 396 x^{5} + 612 x^{6} - 792 x^{7} + 856 x^{8} - 760 x^{9} + 528 x^{10} - 256 x^{11} + 64 x^{12}$ | |
Frobenius angles: | $\pm0.0435981566527$, $\pm0.0833333333333$, $\pm0.250000000000$, $\pm0.329312442367$, $\pm0.527830414776$, $\pm0.583333333333$ |
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: | $3$ |
Slopes: | $[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1$ | $4615$ | $136825$ | $8699275$ | $1387658981$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-5$ | $7$ | $4$ | $7$ | $40$ | $52$ | $58$ | $231$ | $607$ | $1092$ |
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.ac_c $\times$ 3.2.ae_j_ap 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.bqdecdesiy 3 . 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.a_ae $\times$ 3.4.c_ad_an. 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.ae 2 $\times$ 1.8.e $\times$ 3.8.ab_ag_bb. The endomorphism algebra for each factor is: - 1.8.ae 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 1.8.e : \(\Q(\sqrt{-1}) \).
- 3.8.ab_ag_bb : \(\Q(\zeta_{7})\).
- Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ae 2 $\times$ 1.16.i $\times$ 3.16.ak_bl_adt. The endomorphism algebra for each factor is: - 1.16.ae 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 3.16.ak_bl_adt : \(\Q(\zeta_{7})\).
- Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.a 3 $\times$ 3.64.an_ag_bcp. The endomorphism algebra for each factor is: - 1.64.a 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
- 3.64.an_ag_bcp : \(\Q(\zeta_{7})\).
- Endomorphism algebra over $\F_{2^{7}}$
The base change of $A$ to $\F_{2^{7}}$ is 1.128.aq $\times$ 1.128.an 3 $\times$ 2.128.aq_ey. The endomorphism algebra for each factor is: - 1.128.aq : \(\Q(\sqrt{-1}) \).
- 1.128.an 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
- 2.128.aq_ey : \(\Q(\zeta_{12})\).
- Endomorphism algebra over $\F_{2^{12}}$
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey 3 $\times$ 3.4096.agz_bbmw_adccgn. The endomorphism algebra for each factor is: - 1.4096.ey 3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 3.4096.agz_bbmw_adccgn : \(\Q(\zeta_{7})\).
- 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.a_ayge. 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.a_ayge : \(\Q(\zeta_{12})\).
- Endomorphism algebra over $\F_{2^{21}}$
The base change of $A$ to $\F_{2^{21}}$ is 1.2097152.adau 2 $\times$ 1.2097152.dau $\times$ 1.2097152.edn 3 . The endomorphism algebra for each factor is: - 1.2097152.adau 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 1.2097152.dau : \(\Q(\sqrt{-1}) \).
- 1.2097152.edn 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
- Endomorphism algebra over $\F_{2^{28}}$
The base change of $A$ to $\F_{2^{28}}$ is 1.268435456.ayge 2 $\times$ 1.268435456.blhf 3 $\times$ 1.268435456.bwmi. The endomorphism algebra for each factor is: - 1.268435456.ayge 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 1.268435456.blhf 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
- 1.268435456.bwmi : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- Endomorphism algebra over $\F_{2^{42}}$
The base change of $A$ to $\F_{2^{42}}$ is 1.4398046511104.ahxvrd 3 $\times$ 1.4398046511104.a 3 . The endomorphism algebra for each factor is: - 1.4398046511104.ahxvrd 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
- 1.4398046511104.a 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
Base change
This is a primitive isogeny class.