Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - x + 8 x^{2} )( 1 - 5 x + 8 x^{2} )^{2}$ |
$1 - 11 x + 59 x^{2} - 201 x^{3} + 472 x^{4} - 704 x^{5} + 512 x^{6}$ | |
Frobenius angles: | $\pm0.154919815756$, $\pm0.154919815756$, $\pm0.443432958871$ |
Angle rank: | $2$ (numerical) |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $3$ |
Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $128$ | $250880$ | $138322304$ | $68690944000$ | $35473840048768$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-2$ | $62$ | $526$ | $4094$ | $33038$ | $264638$ | $2105486$ | $16787966$ | $134209678$ | $1073708222$ |
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 $\F_{2^{3}}$.
Endomorphism algebra over $\F_{2^{3}}$The isogeny class factors as 1.8.af 2 $\times$ 1.8.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.