Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 23 x^{2} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.333333333333$, $\pm0.666666666667$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{23})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $35$ |
| Cyclic group of points: | yes |
This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $553$ | $305809$ | $148011556$ | $78607897641$ | $41426517649993$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $24$ | $576$ | $12168$ | $280900$ | $6436344$ | $147987222$ | $3404825448$ | $78311544964$ | $1801152661464$ | $41426524086336$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 35 curves (of which all are hyperelliptic):
- $y^2=4 x^6+9 x^5+21 x^4+18 x^3+22 x^2+2 x+13$
- $y^2=20 x^6+22 x^5+13 x^4+21 x^3+18 x^2+10 x+19$
- $y^2=15 x^6+17 x^5+3 x^4+2 x^3+5 x^2+21 x+10$
- $y^2=6 x^6+16 x^5+15 x^4+10 x^3+2 x^2+13 x+4$
- $y^2=14 x^6+3 x^5+x^4+15 x^3+11 x^2+12 x+2$
- $y^2=x^6+15 x^5+5 x^4+6 x^3+9 x^2+14 x+10$
- $y^2=4 x^6+22 x^4+17 x^3+20 x^2+19 x+2$
- $y^2=20 x^6+18 x^4+16 x^3+8 x^2+3 x+10$
- $y^2=4 x^6+19 x^5+19 x^4+4 x^3+2 x^2+7 x+11$
- $y^2=20 x^6+3 x^5+3 x^4+20 x^3+10 x^2+12 x+9$
- $y^2=21 x^6+4 x^5+9 x^4+22 x^3+5 x^2+16 x+3$
- $y^2=13 x^6+20 x^5+22 x^4+18 x^3+2 x^2+11 x+15$
- $y^2=7 x^6+13 x^5+11 x^4+14 x^3+20 x^2+13 x+6$
- $y^2=12 x^6+19 x^5+9 x^4+x^3+8 x^2+19 x+7$
- $y^2=19 x^6+21 x^5+20 x^4+3 x^3+4 x^2+15 x+8$
- $y^2=3 x^6+13 x^5+8 x^4+15 x^3+20 x^2+6 x+17$
- $y^2=21 x^6+15 x^5+20 x^4+8 x^3+12 x^2+x+18$
- $y^2=13 x^6+6 x^5+8 x^4+17 x^3+14 x^2+5 x+21$
- $y^2=5 x^6+3 x^5+18 x^4+18 x^3+2 x^2+13 x+11$
- $y^2=2 x^6+15 x^5+21 x^4+21 x^3+10 x^2+19 x+9$
- and 15 more
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23^{6}}$.
Endomorphism algebra over $\F_{23}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{23})\). |
| The base change of $A$ to $\F_{23^{6}}$ is 1.148035889.abjzy 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $23$ and $\infty$. |
- Endomorphism algebra over $\F_{23^{2}}$
The base change of $A$ to $\F_{23^{2}}$ is 1.529.x 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ - Endomorphism algebra over $\F_{23^{3}}$
The base change of $A$ to $\F_{23^{3}}$ is the simple isogeny class 2.12167.a_abjzy and its endomorphism algebra is the quaternion algebra over \(\Q(\sqrt{23}) \) ramified at both real infinite places.
Base change
This is a primitive isogeny class.