Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 13 x + 85 x^{2} - 299 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.166921904620$, $\pm0.337099187115$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.272597.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $7$ |
| Isomorphism classes: | 7 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $303$ | $280881$ | $150749469$ | $78596402301$ | $41436887174448$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $11$ | $531$ | $12389$ | $280859$ | $6437956$ | $148035951$ | $3404876395$ | $78311586403$ | $1801155311033$ | $41426509763646$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which all are hyperelliptic):
- $y^2=15 x^6+7 x^5+12 x^3+19 x^2+9 x+17$
- $y^2=14 x^6+4 x^5+x^4+18 x^3+14 x^2+21 x+11$
- $y^2=11 x^6+18 x^5+10 x^4+8 x^3+2 x^2+21 x+20$
- $y^2=10 x^6+14 x^5+14 x^4+18 x^3+15 x^2+6 x+21$
- $y^2=11 x^6+12 x^5+x^4+9 x^3+7 x^2+2 x+21$
- $y^2=10 x^6+8 x^5+14 x^4+18 x^3+9 x^2+21 x+6$
- $y^2=10 x^6+15 x^5+3 x^4+10 x^3+5 x^2+x+9$
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 4.0.272597.1. |
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.n_dh | $2$ | (not in LMFDB) |