Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 8 x + 60 x^{2} + 184 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.586887137846$, $\pm0.690919720514$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.342272.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 8 |
| 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: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $782$ | $311236$ | $143519678$ | $78392878736$ | $41463137593342$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $32$ | $586$ | $11792$ | $280134$ | $6442032$ | $148012426$ | $3404817056$ | $78311259198$ | $1801152282080$ | $41426511770346$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=9 x^6+16 x^5+9 x^4+6 x^3+9 x^2+11 x+3$
- $y^2=2 x^6+10 x^5+11 x^4+19 x^3+4 x^2+16 x+12$
- $y^2=16 x^6+12 x^5+17 x^4+16 x^3+7 x^2+8 x+17$
- $y^2=12 x^6+19 x^5+20 x^4+16 x^3+15 x^2+9 x+19$
- $y^2=9 x^6+22 x^5+11 x^4+x^3+6 x^2+x+21$
- $y^2=6 x^6+16 x^5+5 x^4+19 x^2+21 x+3$
- $y^2=15 x^6+7 x^5+6 x^4+11 x^3+3 x^2+10 x$
- $y^2=9 x^6+7 x^5+4 x^4+17 x^3+3 x^2+20 x+6$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23}$.
Endomorphism algebra over $\F_{23}$| The endomorphism algebra of this simple isogeny class is 4.0.342272.1. |
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 |
|---|---|---|
| 2.23.ai_ci | $2$ | (not in LMFDB) |