Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 2 x )^{4}( 1 - 3 x + 9 x^{2} - 12 x^{3} + 16 x^{4} )$ |
$1 - 11 x + 57 x^{2} - 188 x^{3} + 440 x^{4} - 752 x^{5} + 912 x^{6} - 704 x^{7} + 256 x^{8}$ | |
Frobenius angles: | $0$, $0$, $0$, $0$, $\pm0.272875599394$, $\pm0.469557725221$ |
Angle rank: | $2$ (numerical) |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $11$ | $36531$ | $13099856$ | $3310621875$ | $966715000691$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-6$ | $10$ | $51$ | $194$ | $894$ | $3895$ | $15786$ | $63714$ | $259179$ | $1047010$ |
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 2 $\times$ 2.4.ad_j 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.