Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 11 x + 68 x^{2} - 253 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.162255762163$, $\pm0.411666868096$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-214 -22 \sqrt{33}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 6 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $334$ | $287908$ | $149932600$ | $78313279264$ | $41421476307754$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $13$ | $545$ | $12322$ | $279849$ | $6435563$ | $148057850$ | $3405040421$ | $78311655409$ | $1801151383726$ | $41426494361225$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=19 x^6+9 x^5+17 x^4+18 x^3+15 x^2+18 x+11$
- $y^2=14 x^6+20 x^4+5 x^3+9 x^2+5 x+13$
- $y^2=5 x^6+4 x^5+17 x^4+22 x^3+6 x^2+7 x+22$
- $y^2=17 x^6+6 x^5+17 x^4+5 x^3+x+21$
- $y^2=10 x^6+4 x^5+14 x^4+2 x^3+14 x^2+16 x+14$
- $y^2=6 x^6+22 x^5+18 x^4+13 x^3+11 x^2+7 x+22$
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 \(\Q(\sqrt{-214 -22 \sqrt{33}})\). |
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.l_cq | $2$ | (not in LMFDB) |