Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 2 x )^{6}( 1 - x + 4 x^{2} )$ |
$1 - 13 x + 76 x^{2} - 268 x^{3} + 640 x^{4} - 1072 x^{5} + 1216 x^{6} - 832 x^{7} + 256 x^{8}$ | |
Frobenius angles: | $0$, $0$, $0$, $0$, $0$, $0$, $\pm0.419569376745$ |
Angle rank: | $1$ (numerical) |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $4$ | $17496$ | $8941324$ | $2733750000$ | $855553548484$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-8$ | $0$ | $28$ | $144$ | $772$ | $3720$ | $15868$ | $64224$ | $258292$ | $1040760$ |
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 $\F_{2^{2}}$.
Endomorphism algebra over $\F_{2^{2}}$The isogeny class factors as 1.4.ae 3 $\times$ 1.4.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.