Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $4$ |
| L-polynomial: | $( 1 - 2 x )^{2}( 1 - x + 4 x^{2} )( 1 - 5 x + 13 x^{2} - 20 x^{3} + 16 x^{4} )$ |
| $1 - 10 x + 50 x^{2} - 165 x^{3} + 388 x^{4} - 660 x^{5} + 800 x^{6} - 640 x^{7} + 256 x^{8}$ | |
| Frobenius angles: | $0$, $0$, $\pm0.140237960897$, $\pm0.387712212190$, $\pm0.419569376745$ |
| Angle rank: | $3$ (numerical) |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
| $p$-rank: | $3$ |
| Slopes: | $[0, 0, 0, 1/2, 1/2, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $20$ | $59400$ | $17949680$ | $3578850000$ | $947363890500$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-5$ | $17$ | $70$ | $209$ | $875$ | $4022$ | $16795$ | $66209$ | $261070$ | $1043577$ |
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 $\times$ 1.4.ab $\times$ 2.4.af_n 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.