Invariants
Base field: | $\F_{2}$ |
Dimension: | $5$ |
L-polynomial: | $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 3 x + 2 x^{2} + x^{3} + 4 x^{4} - 12 x^{5} + 8 x^{6} )$ |
$1 - 7 x + 22 x^{2} - 39 x^{3} + 44 x^{4} - 48 x^{5} + 88 x^{6} - 156 x^{7} + 176 x^{8} - 112 x^{9} + 32 x^{10}$ | |
Frobenius angles: | $\pm0.0992589862044$, $\pm0.186455299510$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.757883870938$ |
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: | $3$ |
Slopes: | $[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1$ | $725$ | $50869$ | $5093125$ | $58601341$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-4$ | $0$ | $11$ | $44$ | $51$ | $87$ | $136$ | $188$ | $506$ | $1015$ |
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 $\times$ 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 3 $\times$ 1.268435456.bwmi 2 . The endomorphism algebra for each factor is:
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- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.a 2 $\times$ 3.4.af_s_abp. The endomorphism algebra for each factor is: - 1.4.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 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.i 2 $\times$ 3.16.l_cg_jf. The endomorphism algebra for each factor is: - 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 3 . The endomorphism algebra for each factor is: - 1.128.aq 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 1.128.n 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
- Endomorphism algebra over $\F_{2^{14}}$
The base change of $A$ to $\F_{2^{14}}$ is 1.16384.a 2 $\times$ 1.16384.dj 3 . The endomorphism algebra for each factor is: - 1.16384.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 1.16384.dj 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
Base change
This is a primitive isogeny class.