Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 5 x + 8 x^{2} )( 1 - 4 x + 8 x^{2} )^{2}$ |
$1 - 13 x + 80 x^{2} - 288 x^{3} + 640 x^{4} - 832 x^{5} + 512 x^{6}$ | |
Frobenius angles: | $\pm0.154919815756$, $\pm0.250000000000$, $\pm0.250000000000$ |
Angle rank: | $1$ (numerical) |
This isogeny class is not simple, not primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $100$ | $236600$ | $150888700$ | $73972990000$ | $36039459252500$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-4$ | $56$ | $572$ | $4400$ | $33556$ | $263144$ | $2095852$ | $16766816$ | $134192516$ | $1073731736$ |
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 $\F_{2^{12}}$.
Endomorphism algebra over $\F_{2^{3}}$The isogeny class factors as 1.8.af $\times$ 1.8.ae 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 $\F_{2^{12}}$ is 1.4096.bv $\times$ 1.4096.ey 2 . The endomorphism algebra for each factor is:
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- Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj $\times$ 1.64.a 2 . The endomorphism algebra for each factor is: - 1.64.aj : \(\Q(\sqrt{-7}) \).
- 1.64.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{3}}$.
Subfield | Primitive Model |
$\F_{2}$ | 3.2.ab_c_ag |
$\F_{2}$ | 3.2.f_o_y |