Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 11 x + 63 x^{2} - 253 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.0980658819420$, $\pm0.437882154474$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-194 -22 \sqrt{53}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $10$ |
| Isomorphism classes: | 10 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $329$ | $281953$ | $147910175$ | $78003143309$ | $41398488019504$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $13$ | $535$ | $12157$ | $278739$ | $6431988$ | $148052035$ | $3405004231$ | $78311416179$ | $1801152527011$ | $41426519279550$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=7 x^6+17 x^5+19 x^4+3 x^3+11 x^2+x+11$
- $y^2=21 x^6+10 x^5+21 x^4+19 x^3+22 x^2+13 x+9$
- $y^2=11 x^6+11 x^5+11 x^4+22 x^3+15 x^2+8 x+3$
- $y^2=18 x^6+19 x^5+7 x^4+9 x^3+13 x^2+2 x$
- $y^2=19 x^6+12 x^5+22 x^4+11 x^3+2 x^2+21 x+10$
- $y^2=17 x^6+x^5+9 x^4+5 x^3+21 x^2+21 x+7$
- $y^2=10 x^6+19 x^5+2 x^4+16 x^3+8 x^2+18 x+4$
- $y^2=5 x^6+3 x^5+11 x^4+3 x^3+6 x^2+21 x+7$
- $y^2=8 x^5+8 x^4+2 x^3+3 x^2+12 x+12$
- $y^2=19 x^6+21 x^5+20 x^4+5 x^3+12 x^2+5 x+17$
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 \(\Q(\sqrt{-194 -22 \sqrt{53}})\). |
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_cl | $2$ | (not in LMFDB) |