Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 12 x + 76 x^{2} - 276 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.156924629149$, $\pm0.379300706967$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-50 +12 \sqrt{6}})\) |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $318$ | $284292$ | $150246414$ | $78411145104$ | $41421021299598$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $538$ | $12348$ | $280198$ | $6435492$ | $148044058$ | $3404991252$ | $78311930686$ | $1801154110284$ | $41426498189818$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=19 x^6+18 x^5+9 x^4+17 x^3+13 x^2+7$
- $y^2=22 x^5+9 x^4+7 x^2+22 x+5$
- $y^2=21 x^6+7 x^4+12 x^3+4 x^2+3 x+20$
- $y^2=5 x^5+6 x^4+8 x^3+22 x^2+22 x+2$
- $y^2=17 x^6+21 x^5+21 x^2+15 x+10$
- $y^2=14 x^6+9 x^5+14 x^4+17 x^3+18 x^2+18 x+17$
- $y^2=11 x^6+17 x^5+7 x^4+19 x^3+21 x^2+3 x+5$
- $y^2=4 x^6+5 x^5+22 x^4+15 x^3+x^2+22 x+22$
- $y^2=10 x^6+12 x^5+12 x^4+3 x^3+8 x^2+14 x+20$
- $y^2=13 x^6+7 x^5+3 x^4+22 x^3+17 x^2+20 x+22$
- $y^2=5 x^6+17 x^4+14 x^3+19 x^2+x+17$
- $y^2=8 x^6+16 x^5+5 x^4+16 x^3+17 x^2+22$
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 \(\Q(\sqrt{-50 +12 \sqrt{6}})\). |
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.m_cy | $2$ | (not in LMFDB) |