Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $4$ |
| L-polynomial: | $( 1 - 2 x )^{4}( 1 - 2 x + x^{2} - 8 x^{3} + 16 x^{4} )$ |
| $1 - 10 x + 41 x^{2} - 96 x^{3} + 184 x^{4} - 384 x^{5} + 656 x^{6} - 640 x^{7} + 256 x^{8}$ | |
| Frobenius angles: | $0$, $0$, $0$, $0$, $\pm0.0935673124239$, $\pm0.651114279890$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $8$ | $18144$ | $6473096$ | $3311280000$ | $1005980343048$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-5$ | $-1$ | $7$ | $191$ | $935$ | $3743$ | $16039$ | $65279$ | $260071$ | $1046559$ |
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.ac_b 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.