Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 11 x + 65 x^{2} - 253 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.125643553687$, $\pm0.428178140928$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.29725.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 6 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $331$ | $284329$ | $148717969$ | $78130481581$ | $41409806217616$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $13$ | $539$ | $12223$ | $279195$ | $6433748$ | $148058543$ | $3405050585$ | $78311674819$ | $1801152716959$ | $41426511980414$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=14 x^6+2 x^5+x^4+17 x^3+15 x^2+9 x+22$
- $y^2=15 x^6+20 x^5+20 x^4+19 x^3+3 x^2+18 x+7$
- $y^2=20 x^6+20 x^5+15 x^4+3 x^3+x^2+10 x+21$
- $y^2=21 x^6+20 x^4+18 x^3+2 x^2+21 x+17$
- $y^2=7 x^6+22 x^5+14 x^4+7 x^3+5 x^2+8 x+9$
- $y^2=10 x^6+13 x^5+2 x^4+10 x^3+20 x^2+14 x+17$
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 4.0.29725.1. |
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.l_cn | $2$ | (not in LMFDB) |