Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $4$ |
| L-polynomial: | $1 + x + 3 x^{2} + 5 x^{3} + 16 x^{4} + 15 x^{5} + 27 x^{6} + 27 x^{7} + 81 x^{8}$ |
| Frobenius angles: | $\pm0.264830336654$, $\pm0.372732979144$, $\pm0.664830336654$, $\pm0.827267020856$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 8.0.58140625.2 |
| Galois group: | $C_2^2:C_4$ |
| Jacobians: | $7$ |
| Isomorphism classes: | 102 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $4$ |
| Slopes: | $[0, 0, 0, 0, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $176$ | $14080$ | $660176$ | $68710400$ | $3152148736$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $5$ | $15$ | $35$ | $119$ | $220$ | $675$ | $2105$ | $6719$ | $19475$ | $58850$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which 1 is hyperelliptic):
- $y^2=x^{10}+x^9+2 x^5+2 x^2+x$
- $x y+t^2=x^2 z+y^2 z+x z^2-z^3+x y t-y^2 t+y z t-z^2 t=0$
- $x t-y z=y^3+x^2 z-z^3+x^2 t+y^2 t-x z t-z^2 t-x t^2-z t^2+t^3=0$
- $x t-y z=x y^2+x^2 z+z^3+x y t+y^2 t-x z t+z^2 t+x t^2+y t^2-z t^2+t^3=0$
- $x t-y z=x y^2+x^2 z-z^3+x^2 t+y^2 t-x z t-z^2 t+t^3=0$
- $x t-y z=x^2 y+y^3+x^2 z-z^3+x y t+y^2 t-x z t+z^2 t+x t^2-y t^2+z t^2+t^3=0$
- $x^2+y^2+z t=y^3+y^2 z+y z^2+x y t-z^2 t-y t^2=0$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{5}}$.
Endomorphism algebra over $\F_{3}$| The endomorphism algebra of this simple isogeny class is 8.0.58140625.2. |
| The base change of $A$ to $\F_{3^{5}}$ is 2.243.am_w 2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.1525.1$)$ |
Base change
This is a primitive isogeny class.