Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $6$ |
| L-polynomial: | $( 1 + 2 x^{2} )( 1 - x + 2 x^{2} )^{3}( 1 - x + 3 x^{2} - 2 x^{3} + 4 x^{4} )$ |
| $1 - 4 x + 17 x^{2} - 41 x^{3} + 98 x^{4} - 165 x^{5} + 272 x^{6} - 330 x^{7} + 392 x^{8} - 328 x^{9} + 272 x^{10} - 128 x^{11} + 64 x^{12}$ | |
| Frobenius angles: | $\pm0.306143893905$, $\pm0.384973271919$, $\pm0.384973271919$, $\pm0.384973271919$, $\pm0.5$, $\pm0.570118980449$ |
| Angle rank: | $3$ (numerical) |
| Jacobians: | $0$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
| $p$-rank: | $5$ |
| Slopes: | $[0, 0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $120$ | $253440$ | $1975680$ | $10137600$ | $483153000$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-1$ | $23$ | $26$ | $7$ | $9$ | $44$ | $125$ | $319$ | $602$ | $943$ |
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 $\F_{2}$The isogeny class factors as 1.2.ab 3 $\times$ 1.2.a $\times$ 2.2.ab_d and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
|
The base change of $A$ to $\F_{2^{2}}$ is 1.4.d 3 $\times$ 1.4.e $\times$ 2.4.f_n. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.