Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 2 x )^{6}( 1 - 3 x + 4 x^{2} )$ |
$1 - 15 x + 100 x^{2} - 388 x^{3} + 960 x^{4} - 1552 x^{5} + 1600 x^{6} - 960 x^{7} + 256 x^{8}$ | |
Frobenius angles: | $0$, $0$, $0$, $0$, $0$, $0$, $\pm0.230053456163$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2$ | $11664$ | $8706026$ | $3280500000$ | $960278982842$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-10$ | $-8$ | $26$ | $192$ | $890$ | $3760$ | $15530$ | $63552$ | $258074$ | $1041232$ |
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.ad 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.