Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 12 x + 71 x^{2} - 276 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.0764113711973$, $\pm0.409744704531$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{11})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
| Cyclic group of points: | yes |
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: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $313$ | $278257$ | $148032724$ | $78016862889$ | $41382431470873$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $528$ | $12168$ | $278788$ | $6429492$ | $148029558$ | $3404945052$ | $78311536516$ | $1801152661464$ | $41426508436368$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=5 x^6+7 x^5+8 x^4+15 x^3+3 x^2+11 x+11$
- $y^2=18 x^6+10 x^5+x^4+17 x^3+15 x^2+18 x+20$
- $y^2=15 x^6+19 x^5+8 x^4+2 x^3+2 x+15$
- $y^2=20 x^6+3 x^5+22 x^4+21 x^3+3 x^2+4 x+5$
- $y^2=17 x^6+21 x^5+6 x^4+8 x^3+18 x^2+12 x+17$
- $y^2=10 x^6+4 x^5+10 x^4+x^3+5 x^2+4 x+17$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23^{6}}$.
Endomorphism algebra over $\F_{23}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{11})\). |
| The base change of $A$ to $\F_{23^{6}}$ is 1.148035889.aeru 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-33}) \)$)$ |
- Endomorphism algebra over $\F_{23^{2}}$
The base change of $A$ to $\F_{23^{2}}$ is the simple isogeny class 2.529.ac_auf and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{11})\). - Endomorphism algebra over $\F_{23^{3}}$
The base change of $A$ to $\F_{23^{3}}$ is the simple isogeny class 2.12167.a_aeru and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{11})\).
Base change
This is a primitive isogeny class.