Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 3 x + 15 x^{2} - 29 x^{3} + 75 x^{4} - 75 x^{5} + 125 x^{6}$ |
| Frobenius angles: | $\pm0.277336024600$, $\pm0.453376467205$, $\pm0.537962020378$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.29649159.1 |
| Galois group: | $A_4\times C_2$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 5 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $109$ | $35207$ | $2327041$ | $228739879$ | $30794411969$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $47$ | $147$ | $587$ | $3153$ | $15851$ | $77640$ | $388451$ | $1952841$ | $9773327$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 0 are hyperelliptic):
- $x^4+x^3 y+x^3 z+x z^3+2 y^4+y^2 z^2=0$
- $2 x^4+2 x^3 y+4 x^3 z+4 x^2 y z+4 x^2 z^2+x y^3+x z^3+2 y^4+y^2 z^2=0$
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 6.0.29649159.1. |
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_p_bd | $2$ | 3.25.v_ht_bun |