Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 4 x + 17 x^{2} - 38 x^{3} + 85 x^{4} - 100 x^{5} + 125 x^{6}$ |
| Frobenius angles: | $\pm0.249213002135$, $\pm0.405288933323$, $\pm0.534316041759$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.78766784.1 |
| Galois group: | $S_4\times C_2$ |
| Jacobians: | $2$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $86$ | $31820$ | $2415998$ | $238395440$ | $30766936966$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $2$ | $44$ | $152$ | $612$ | $3152$ | $15776$ | $77842$ | $389468$ | $1952162$ | $9762264$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 1 is hyperelliptic):
- $y^2=x^8+x^6+x^5+3 x^4+2 x^3+2 x+2$
- $3 x^4+3 x^3 y+4 x^3 z+x^2 y z+x^2 z^2+x z^3+2 y^4+y^2 z^2=0$
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 6.0.78766784.1. |
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.e_r_bm | $2$ | 3.25.s_fz_bim |