Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 12 x + 79 x^{2} - 276 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.201562474799$, $\pm0.353216304931$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.342288.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 8 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $321$ | $287937$ | $151580052$ | $78634730889$ | $41434923729201$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $544$ | $12456$ | $280996$ | $6437652$ | $148032502$ | $3404847612$ | $78311269444$ | $1801152693432$ | $41426497057984$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=7 x^6+19 x^5+16 x^4+11 x^3+12 x^2+6 x+22$
- $y^2=19 x^6+20 x^5+11 x^4+3 x^3+17 x^2+4$
- $y^2=8 x^6+18 x^5+3 x^4+6 x^3+18 x^2+3 x+20$
- $y^2=12 x^6+5 x^5+18 x^4+19 x^3+3 x^2+5 x+18$
- $y^2=5 x^6+21 x^5+4 x^4+15 x^3+14 x^2+10 x+14$
- $y^2=14 x^6+14 x^5+16 x^4+14 x^3+9 x^2+11 x+2$
- $y^2=14 x^6+8 x^5+x^4+22 x^3+11 x^2+2 x+16$
- $y^2=10 x^6+3 x^5+20 x^3+21 x^2+21 x+5$
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 4.0.342288.1. |
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.m_db | $2$ | (not in LMFDB) |