Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 2 x + 7 x^{2} + 23 x^{3} + 35 x^{4} + 50 x^{5} + 125 x^{6}$ |
| Frobenius angles: | $\pm0.376847748144$, $\pm0.469230501932$, $\pm0.872660818878$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.1249779231.1 |
| Galois group: | $S_4\times C_2$ |
| Jacobians: | $5$ |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $243$ | $22599$ | $2601072$ | $227278143$ | $28602678528$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $36$ | $161$ | $580$ | $2923$ | $15849$ | $78184$ | $392228$ | $1949417$ | $9768591$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which 0 are hyperelliptic):
- $4 x^3 y+4 x^3 z+x^2 y^2+2 x^2 y z+x^2 z^2+x z^3+y^3 z=0$
- $4 x^3 y+4 x^3 z+x^2 y^2+3 x^2 y z+x^2 z^2+x y z^2+x z^3+y^3 z=0$
- $x^4+2 x^3 z+3 x^2 y^2+3 x^2 y z+x^2 z^2+x y z^2+x z^3+y^3 z=0$
- $3 x^4+4 x^3 y+x^3 z+3 x^2 y^2+2 x^2 y z+3 x^2 z^2+x z^3+y^4+y^2 z^2=0$
- $x^3 y+x^3 z+2 x^2 y^2+2 x^2 y z+4 x^2 z^2+x z^3+y^4+y^2 z^2=0$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5}$.
Endomorphism algebra over $\F_{5}$| The endomorphism algebra of this simple isogeny class is 6.0.1249779231.1. |
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 |
|---|---|---|
| 3.5.ac_h_ax | $2$ | 3.25.k_bb_l |