Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 10 x + 65 x^{2} + 230 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.585672095708$, $\pm0.783089164485$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-61 -10 \sqrt{6}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $12$ |
| Isomorphism classes: | 12 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $835$ | $296425$ | $144912580$ | $78441465625$ | $41426821687675$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $34$ | $560$ | $11908$ | $280308$ | $6436394$ | $148049030$ | $3404721158$ | $78310916068$ | $1801157083804$ | $41426488524800$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=3 x^6+21 x^5+11 x^4+21 x^3+15 x^2+8 x+3$
- $y^2=3 x^6+19 x^5+7 x^4+14 x^3+2 x^2+3 x+9$
- $y^2=4 x^6+8 x^5+4 x^4+3 x^3+12 x^2+20 x+1$
- $y^2=2 x^6+10 x^5+8 x^4+8 x^3+8 x^2+10 x+18$
- $y^2=9 x^6+10 x^5+18 x^4+19 x^3+9 x^2+1$
- $y^2=16 x^6+16 x^5+19 x^4+12 x^3+15 x^2+17$
- $y^2=7 x^6+18 x^5+20 x^4+12 x^3+9 x^2+19 x+16$
- $y^2=17 x^6+17 x^5+17 x^4+15 x^3+14 x^2+18 x+12$
- $y^2=9 x^6+11 x^5+x^4+20 x^3+13 x^2+2$
- $y^2=20 x^6+6 x^5+15 x^4+5 x^3+6 x^2+12 x+18$
- $y^2=16 x^6+11 x^5+12 x^4+19 x^3+13 x^2+19 x+15$
- $y^2=12 x^6+5 x^4+14 x^3+9 x^2+12 x+6$
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 \(\Q(\sqrt{-61 -10 \sqrt{6}})\). |
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.ak_cn | $2$ | (not in LMFDB) |