Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $5$ |
| L-polynomial: | $( 1 + 2 x^{2} )( 1 + x + 2 x^{2} )( 1 + 3 x + 5 x^{2} + 7 x^{3} + 10 x^{4} + 12 x^{5} + 8 x^{6} )$ |
| $1 + 4 x + 12 x^{2} + 26 x^{3} + 47 x^{4} + 72 x^{5} + 94 x^{6} + 104 x^{7} + 96 x^{8} + 64 x^{9} + 32 x^{10}$ | |
| Frobenius angles: | $\pm0.358750840369$, $\pm0.5$, $\pm0.615026728081$, $\pm0.683161207432$, $\pm0.894721499061$ |
| Angle rank: | $4$ (numerical) |
| Jacobians: | $1$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $4$ |
| Slopes: | $[0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $552$ | $6624$ | $21528$ | $900864$ | $40141992$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $13$ | $7$ | $13$ | $37$ | $49$ | $119$ | $317$ | $457$ | $1133$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is not hyperelliptic):
- $x t + y^2 + y z + z^2 + z t = x t + x u + y^2 + y z + y t + y u + z^2 + z t + t u + u^2 = x y + x u + y t + y u + z^2 + z t + t u + u^2 = 0$
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.a $\times$ 1.2.b $\times$ 3.2.d_f_h 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 $\times$ 1.4.e $\times$ 3.4.b_d_af. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.