Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $4$ |
| L-polynomial: | $( 1 - 2 x )^{4}( 1 - 2 x + 3 x^{2} - 8 x^{3} + 16 x^{4} )$ |
| $1 - 10 x + 43 x^{2} - 112 x^{3} + 232 x^{4} - 448 x^{5} + 688 x^{6} - 640 x^{7} + 256 x^{8}$ | |
| Frobenius angles: | $0$, $0$, $0$, $0$, $\pm0.168977707736$, $\pm0.618033150523$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $10$ | $24300$ | $7947310$ | $3523500000$ | $1085183351050$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-5$ | $3$ | $19$ | $207$ | $1015$ | $3891$ | $15955$ | $65247$ | $259831$ | $1041603$ |
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_d 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.