Invariants
Base field: | $\F_{2}$ |
Dimension: | $5$ |
L-polynomial: | $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 3 x + 5 x^{2} - 7 x^{3} + 10 x^{4} - 12 x^{5} + 8 x^{6} )$ |
$1 - 7 x + 25 x^{2} - 59 x^{3} + 106 x^{4} - 160 x^{5} + 212 x^{6} - 236 x^{7} + 200 x^{8} - 112 x^{9} + 32 x^{10}$ | |
Frobenius angles: | $\pm0.105278500939$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.316838792568$, $\pm0.641249159631$ |
Angle rank: | $3$ (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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2$ | $2300$ | $51038$ | $3910000$ | $72017402$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-4$ | $6$ | $14$ | $38$ | $56$ | $42$ | $94$ | $254$ | $500$ | $1126$ |
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 $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ac 2 $\times$ 3.2.ad_f_ah 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 $\F_{2^{4}}$ is 1.16.i 2 $\times$ 3.16.f_br_fn. 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.b_d_af. The endomorphism algebra for each factor is: - 1.4.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 3.4.b_d_af : 6.0.679024.1.
Base change
This is a primitive isogeny class.