Invariants
Base field: | $\F_{2}$ |
Dimension: | $5$ |
L-polynomial: | $( 1 - x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )^{2}$ |
$1 - 7 x + 27 x^{2} - 73 x^{3} + 149 x^{4} - 237 x^{5} + 298 x^{6} - 292 x^{7} + 216 x^{8} - 112 x^{9} + 32 x^{10}$ | |
Frobenius angles: | $\pm0.123548644961$, $\pm0.123548644961$, $\pm0.384973271919$, $\pm0.456881978294$, $\pm0.456881978294$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $0$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2$ | $2888$ | $80864$ | $467856$ | $20317462$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-4$ | $10$ | $14$ | $2$ | $16$ | $100$ | $220$ | $322$ | $518$ | $1090$ |
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 $\F_{2^{6}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ab $\times$ 2.2.ad_f 2 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 $\F_{2^{6}}$ is 1.64.aj $\times$ 1.64.l 4 . 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.d $\times$ 2.4.b_ad 2 . The endomorphism algebra for each factor is: - 1.4.d : \(\Q(\sqrt{-7}) \).
- 2.4.b_ad 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{5})\)$)$
- Endomorphism algebra over $\F_{2^{3}}$
The base change of $A$ to $\F_{2^{3}}$ is 1.8.f $\times$ 2.8.a_l 2 . The endomorphism algebra for each factor is: - 1.8.f : \(\Q(\sqrt{-7}) \).
- 2.8.a_l 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{5})\)$)$
Base change
This is a primitive isogeny class.