Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 3 x + 9 x^{2} - 18 x^{3} + 45 x^{4} - 75 x^{5} + 125 x^{6}$ |
| Frobenius angles: | $\pm0.183865915505$, $\pm0.394525812667$, $\pm0.663830602751$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.623195856.1 |
| Galois group: | $S_4\times C_2$ |
| Jacobians: | $18$ |
| Isomorphism classes: | 40 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $84$ | $23184$ | $1944432$ | $262999296$ | $31064875044$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $35$ | $126$ | $671$ | $3183$ | $15512$ | $79131$ | $392543$ | $1948590$ | $9756635$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which 6 are hyperelliptic):
- $y^2=x^7+x^2+x+4$
- $y^2=x^7+x^4+4 x^3+4 x^2+x+3$
- $y^2=x^7+2 x^4+3 x^3+3 x^2+x+3$
- $y^2=x^7+2 x^4+3 x^3+4 x^2+2$
- $y^2=x^7+x^5+2 x^4+4 x^2+4 x+2$
- $y^2=x^7+2 x^5+2 x^3+x^2+1$
- $3 x^4+4 x^3 y+3 x^3 z+4 x^2 y^2+x^2 y z+x z^3+y^4=0$
- $x^4+2 x^3 y+x^3 z+x^2 y^2+x^2 y z+x y^2 z+x z^3+y^4=0$
- $4 x^4+x^3 y+2 x^3 z+x^2 y^2+x^2 y z+2 x y^2 z+x z^3+y^4=0$
- $x^4+x^3 y+x^3 z+2 x^2 y^2+x^2 y z+3 x^2 z^2+x y z^2+x z^3+y^3 z=0$
- $3 x^4+2 x^3 y+x^2 y^2+3 x^2 y z+4 x^2 z^2+x y z^2+x z^3+y^3 z=0$
- $x^4+x^3 y+x^3 z+3 x^2 y^2+4 x^2 z^2+x z^3+y^4+y^2 z^2=0$
- $2 x^4+2 x^3 y+4 x^3 z+x^2 y^2+x^2 y z+x z^3+2 y^4+y^2 z^2=0$
- $3 x^4+2 x^3 y+4 x^3 z+x^2 y z+x^2 z^2+x z^3+2 y^4+y^2 z^2=0$
- $4 x^4+x^3 y+2 x^3 z+3 x^2 y^2+x^2 y z+x y^3+x z^3+2 y^4+y^2 z^2=0$
- $2 x^4+3 x^3 z+3 x^2 y^2+x^2 y z+x y^3+x z^3+2 y^4+y^2 z^2=0$
- $x^4+4 x^3 y+4 x^3 z+x^2 y^2+3 x^2 y z+x y^3+x z^3+2 y^4+y^2 z^2=0$
- $4 x^3 y+x^3 z+4 x^2 y^2+4 x^2 z^2+x y^3+x z^3+2 y^4+y^2 z^2=0$
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 6.0.623195856.1. |
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_j_s | $2$ | 3.25.j_cl_la |