Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 9 x + 63 x^{2} + 207 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.590734115210$, $\pm0.728221218053$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.748501.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 6 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $809$ | $304993$ | $143819975$ | $78472563949$ | $41444818122704$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $33$ | $575$ | $11817$ | $280419$ | $6439188$ | $148022795$ | $3404830371$ | $78310872019$ | $1801154576871$ | $41426507018750$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=11 x^6+22 x^5+2 x^4+15 x^3+7 x^2+9 x+4$
- $y^2=12 x^6+16 x^5+21 x^4+20 x^3+x^2+2 x+5$
- $y^2=3 x^6+22 x^5+14 x^4+17 x^3+9 x^2+18 x+10$
- $y^2=9 x^6+17 x^4+3 x^3+18 x^2+21 x+19$
- $y^2=3 x^6+10 x^5+x^4+7 x^3+19 x^2+19 x+17$
- $y^2=16 x^6+4 x^5+18 x^4+3 x^3+14 x^2+4 x+13$
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 4.0.748501.1. |
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.aj_cl | $2$ | (not in LMFDB) |