Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 12 x + 70 x^{2} - 276 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.0519709137119$, $\pm0.414830815663$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-8 -2 \sqrt{3}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $14$ |
| Isomorphism classes: | 22 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $312$ | $277056$ | $147591288$ | $77934744576$ | $41372398716792$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $526$ | $12132$ | $278494$ | $6427932$ | $148021486$ | $3404888436$ | $78311140798$ | $1801150592940$ | $41426500509646$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):
- $y^2=5 x^6+20 x^5+12 x^4+x^3+3 x^2+11 x$
- $y^2=11 x^6+22 x^5+16 x^4+x^3+4 x^2+10 x+22$
- $y^2=10 x^6+13 x^5+18 x^4+5 x^3+12 x^2+10 x+11$
- $y^2=21 x^6+11 x^5+7 x^4+16 x^2+21 x+22$
- $y^2=7 x^6+8 x^5+13 x^4+15 x^3+14 x^2+22 x+10$
- $y^2=12 x^6+22 x^5+19 x^4+12 x^3+21 x^2+2 x+5$
- $y^2=10 x^6+11 x^5+14 x^4+19 x^3+9 x^2+13 x+21$
- $y^2=3 x^6+14 x^5+22 x^4+5 x^3+21 x^2+17 x+20$
- $y^2=22 x^6+18 x^5+11 x^4+2 x^3+5 x^2+20 x+14$
- $y^2=20 x^6+19 x^5+2 x^4+4 x^3+14 x^2+16 x+22$
- $y^2=7 x^6+22 x^5+21 x^3+12 x^2+11 x+20$
- $y^2=21 x^6+18 x^5+19 x^4+12 x^3+7 x^2+13 x+10$
- $y^2=13 x^5+x^4+18 x^3+6 x^2+22 x+19$
- $y^2=17 x^6+17 x^5+19 x^4+22 x^3+7 x^2+15 x+18$
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 \(\Q(\sqrt{-8 -2 \sqrt{3}})\). |
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_cs | $2$ | (not in LMFDB) |