Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 7 x + 27 x^{2} - 72 x^{3} + 135 x^{4} - 175 x^{5} + 125 x^{6}$ |
| Frobenius angles: | $\pm0.0767028971338$, $\pm0.313688588913$, $\pm0.486729365003$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.31554496.1 |
| Galois group: | $S_4\times C_2$ |
| Jacobians: | $0$ |
| Isomorphism classes: | 2 |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable.
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $34$ | $18428$ | $2081752$ | $227696368$ | $29704311514$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-1$ | $31$ | $134$ | $583$ | $3039$ | $15604$ | $77895$ | $389919$ | $1956050$ | $9781671$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
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 6.0.31554496.1. |
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.h_bb_cu | $2$ | 3.25.f_aj_adq |