Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + x + 12 x^{2} + 9 x^{3} + 60 x^{4} + 25 x^{5} + 125 x^{6}$ |
| Frobenius angles: | $\pm0.392544792219$, $\pm0.522161388365$, $\pm0.661269982333$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.118697776.1 |
| Galois group: | $S_4\times C_2$ |
| Jacobians: | $4$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $233$ | $37979$ | $1823924$ | $230342635$ | $30430867373$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $49$ | $118$ | $589$ | $3117$ | $15496$ | $78477$ | $391461$ | $1951714$ | $9766509$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which 0 are hyperelliptic):
- $4 x^4+3 x^3 y+4 x^2 y^2+x^2 y z+x y^2 z+x z^3+y^4=0$
- $2 x^4+3 x^3 y+2 x^2 y^2+2 x^2 y z+x y z^2+x z^3+y^3 z=0$
- $3 x^4+3 x^3 y+3 x^3 z+2 x^2 y^2+2 x^2 y z+3 x^2 z^2+x y z^2+x z^3+y^3 z=0$
- $2 x^4+3 x^3 z+3 x^2 y^2+3 x^2 y z+3 x^2 z^2+x y z^2+x z^3+y^3 z=0$
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 6.0.118697776.1. |
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.ab_m_aj | $2$ | 3.25.x_jm_chz |