Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 + 4 x + 8 x^{2} )( 1 + 7 x + 24 x^{2} + 56 x^{3} + 64 x^{4} )$ |
$1 + 11 x + 60 x^{2} + 208 x^{3} + 480 x^{4} + 704 x^{5} + 512 x^{6}$ | |
Frobenius angles: | $\pm0.581839774401$, $\pm0.750000000000$, $\pm0.941488805765$ |
Angle rank: | $2$ (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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1976$ | $256880$ | $127726664$ | $68157970400$ | $35519271367736$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $20$ | $64$ | $488$ | $4064$ | $33080$ | $261712$ | $2097192$ | $16772032$ | $134249240$ | $1073703984$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains no Jacobian of a hyperelliptic curve, but it is unknown whether it contains a Jacobian of a non-hyperelliptic curve.
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.e $\times$ 2.8.h_y 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.ey $\times$ 2.4096.agf_vki. The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.a $\times$ 2.64.ab_adc. The endomorphism algebra for each factor is:
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_a_e |