Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 3 x + 15 x^{2} - 27 x^{3} + 75 x^{4} - 75 x^{5} + 125 x^{6}$ |
| Frobenius angles: | $\pm0.308415961642$, $\pm0.402591248965$, $\pm0.563001890503$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.31171311.1 |
| Galois group: | $A_4\times C_2$ |
| Jacobians: | $3$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $111$ | $35631$ | $2410587$ | $238620807$ | $30192774891$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $47$ | $153$ | $611$ | $3093$ | $15431$ | $77556$ | $391475$ | $1956933$ | $9762947$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which 1 is hyperelliptic):
- $y^2=x^7+4 x^5+2 x^4+4 x^2+x+2$
- $2 x^4+3 x^3 y+2 x^3 z+2 x^2 y^2+3 x^2 y z+3 x^2 z^2+x y z^2+x z^3+y^3 z=0$
- $4 x^3 y+3 x^3 z+2 x^2 y z+x z^3+2 y^4+y^2 z^2=0$
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 6.0.31171311.1. |
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.d_p_bb | $2$ | 3.25.v_if_byv |