Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 5 x + 15 x^{2} + 32 x^{3} + 60 x^{4} + 80 x^{5} + 64 x^{6}$ |
| Frobenius angles: | $\pm0.424864887553$, $\pm0.711106316528$, $\pm0.834728420735$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.7761268.1 |
| Galois group: | $S_4\times C_2$ |
| Jacobians: | $2$ |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $257$ | $5911$ | $247748$ | $18519163$ | $950142107$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $22$ | $61$ | $282$ | $900$ | $4159$ | $16670$ | $65682$ | $260917$ | $1048102$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 0 are hyperelliptic):
- $a x^4+x^3 y+x^3 z+a x y^3+x y^2 z+x y z^2+x z^3+y^2 z^2=0$
- $a^2 x^4+x^3 y+x^3 z+a^2 x y^3+x y^2 z+x y z^2+x z^3+y^2 z^2=0$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{2}}$.
Endomorphism algebra over $\F_{2^{2}}$| The endomorphism algebra of this simple isogeny class is 6.0.7761268.1. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 3.4.af_p_abg | $2$ | 3.16.f_z_ea |