Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 21 x^{2} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.174547532161$, $\pm0.825452467839$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{67})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $9$ |
| Isomorphism classes: | 7 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $509$ | $259081$ | $148059956$ | $78657250681$ | $41426502241589$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $24$ | $488$ | $12168$ | $281076$ | $6436344$ | $148084022$ | $3404825448$ | $78311343268$ | $1801152661464$ | $41426493269528$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=5 x^6+19 x^5+21 x^4+19 x^3+14 x^2+5 x+14$
- $y^2=2 x^6+3 x^5+13 x^4+3 x^3+x^2+2 x+1$
- $y^2=20 x^6+16 x^5+16 x^4+15 x^3+5 x^2+4 x+2$
- $y^2=17 x^6+9 x^5+8 x^4+2 x^3+11 x^2+13 x+15$
- $y^2=9 x^6+21 x^5+8 x^4+3 x^3+12 x^2+19 x+11$
- $y^2=22 x^6+13 x^5+17 x^4+15 x^3+14 x^2+3 x+9$
- $y^2=8 x^6+22 x^5+6 x^3+9 x^2+21 x+2$
- $y^2=12 x^6+17 x^5+6 x^4+20 x^3+4 x^2+18 x+3$
- $y^2=14 x^6+16 x^5+7 x^4+8 x^3+20 x^2+21 x+15$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23^{2}}$.
Endomorphism algebra over $\F_{23}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{67})\). |
| The base change of $A$ to $\F_{23^{2}}$ is 1.529.av 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-67}) \)$)$ |
Base change
This is a primitive isogeny class.