Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 6 x + 23 x^{2} + 138 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.410661054163$, $\pm0.858299445088$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.92736.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $28$ |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $1$ |
| Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $697$ | $285073$ | $150696976$ | $78244841529$ | $41374443782257$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $30$ | $540$ | $12384$ | $279604$ | $6428250$ | $148054830$ | $3404816550$ | $78311845924$ | $1801149370272$ | $41426502374700$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):
- $y^2=6 x^6+15 x^5+8 x^4+9 x^2+20 x+4$
- $y^2=18 x^6+18 x^5+8 x^3+9 x^2+5 x+6$
- $y^2=10 x^6+16 x^4+10 x^3+4 x^2+16 x+21$
- $y^2=7 x^6+11 x^5+8 x^4+2 x^2+20 x+22$
- $y^2=4 x^6+6 x^5+20 x^4+2 x^3+16 x^2+7$
- $y^2=6 x^6+22 x^5+20 x^4+15 x^3+22 x^2+17 x+10$
- $y^2=6 x^6+10 x^5+22 x^4+3 x^3+19 x+18$
- $y^2=10 x^6+15 x^5+17 x^4+13 x^3+13 x^2+15 x+4$
- $y^2=5 x^6+17 x^5+10 x^4+11 x^3+11 x^2+10 x+15$
- $y^2=4 x^6+21 x^5+4 x^4+5 x^3+3 x^2+13 x+2$
- $y^2=6 x^6+22 x^5+8 x^4+17 x^3+x^2+11 x+3$
- $y^2=16 x^6+13 x^5+13 x^4+7 x^3+11 x^2+9 x+14$
- $y^2=12 x^6+8 x^5+5 x^4+7 x^3+16 x+13$
- $y^2=9 x^6+21 x^5+x^4+19 x^3+21 x^2+17 x+11$
- $y^2=x^6+2 x^5+22 x^4+18 x^3+4 x^2+3 x+18$
- $y^2=20 x^6+22 x^5+2 x^4+6 x^3+15 x^2+x+16$
- $y^2=22 x^6+21 x^5+22 x^4+21 x^3+20 x^2+7 x+1$
- $y^2=12 x^6+15 x^5+9 x^4+12 x^3+20 x^2+13 x+4$
- $y^2=19 x^6+11 x^5+x^4+15 x^3+3 x+21$
- $y^2=20 x^6+9 x^5+9 x^4+21 x^3+3 x^2+13 x+21$
- $y^2=20 x^6+6 x^5+11 x^4+4 x^3+10 x^2+18 x+18$
- $y^2=17 x^6+2 x^5+3 x^4+5 x^3+9 x^2+2 x+9$
- $y^2=x^6+13 x^5+4 x^4+16 x^3+5 x^2+11 x+2$
- $y^2=8 x^6+5 x^4+10 x^2+11 x+19$
- $y^2=3 x^6+22 x^5+20 x^4+22 x^3+16 x^2+17 x+4$
- $y^2=13 x^6+13 x^5+8 x^4+x^3+x^2+10 x+8$
- $y^2=14 x^6+7 x^5+9 x^4+3 x^3+16 x+4$
- $y^2=17 x^6+20 x^5+7 x^4+17 x^3+13 x^2+12 x+21$
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 4.0.92736.1. |
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.ag_x | $2$ | (not in LMFDB) |