Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 5 x + 8 x^{2} - 8 x^{3} + 40 x^{4} - 125 x^{5} + 125 x^{6}$ |
| Frobenius angles: | $\pm0.0277077558100$, $\pm0.229975230447$, $\pm0.716266290926$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.23600164.1 |
| Galois group: | $S_4\times C_2$ |
| Jacobians: | $3$ |
| Isomorphism classes: | 4 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $36$ | $11232$ | $1533924$ | $262244736$ | $30222590976$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $1$ | $17$ | $97$ | $673$ | $3096$ | $15353$ | $78037$ | $387745$ | $1947949$ | $9763372$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which 0 are hyperelliptic), and hence is principally polarizable:
- $4x^4+x^3z+x^2y^2+x^2yz+x^2z^2+xy^3+xz^3+2y^4+y^2z^2=0$
- $4x^4+3x^3y+4x^3z+x^2y^2+2x^2yz+2x^2z^2+xy^3+xz^3+2y^4+y^2z^2=0$
- $2x^4+2x^2y^2+3x^2z^2+2xy^3+xz^3+2y^4+y^2z^2=0$
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 6.0.23600164.1. |
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.f_i_i | $2$ | 3.25.aj_cm_aqi |