Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 5 x + 18 x^{2} + 42 x^{3} + 90 x^{4} + 125 x^{5} + 125 x^{6}$ |
| Frobenius angles: | $\pm0.432924113150$, $\pm0.685621524145$, $\pm0.783031703331$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.110965156.1 |
| Galois group: | $S_4\times C_2$ |
| Jacobians: | $5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $406$ | $25172$ | $1676374$ | $262997056$ | $28804926976$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $11$ | $37$ | $107$ | $673$ | $2946$ | $15613$ | $79167$ | $389985$ | $1950839$ | $9762672$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which 2 are hyperelliptic):
- $y^2=x^7+2 x^5+x^4+3 x^3+2 x^2+4 x+1$
- $y^2=x^7+3 x^5+x^4+2 x^3+3 x^2+1$
- $x^4+2 x^3 y+x^2 y^2+x^2 y z+x y z^2+x z^3+y^3 z=0$
- $x^3 y+4 x^3 z+x^2 y^2+x^2 y z+x y z^2+x z^3+y^3 z=0$
- $2 x^4+3 x^3 y+2 x^3 z+x^2 y^2+2 x^2 y z+x^2 z^2+x y z^2+x z^3+y^3 z=0$
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 6.0.110965156.1. |
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.af_s_abq | $2$ | 3.25.l_dg_si |