Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 12 x^{2} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.292004625875$, $\pm0.707995374125$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{34}, \sqrt{-58})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
| Cyclic group of points: | yes |
This isogeny class is simple but not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $542$ | $293764$ | $148018574$ | $78823931536$ | $41426523682382$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $24$ | $554$ | $12168$ | $281670$ | $6436344$ | $148001258$ | $3404825448$ | $78310433854$ | $1801152661464$ | $41426536151114$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=17 x^6+9 x^5+7 x^4+15 x^3+4 x^2+21 x+13$
- $y^2=16 x^6+22 x^5+12 x^4+6 x^3+20 x^2+13 x+19$
- $y^2=6 x^6+10 x^5+21 x^4+11 x^3+14 x^2+10 x$
- $y^2=7 x^6+4 x^5+13 x^4+9 x^3+x^2+4 x$
- $y^2=11 x^6+14 x^5+2 x^4+5 x^3+19 x^2+19 x+16$
- $y^2=9 x^6+x^5+10 x^4+2 x^3+3 x^2+3 x+11$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23^{2}}$.
Endomorphism algebra over $\F_{23}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{34}, \sqrt{-58})\). |
| The base change of $A$ to $\F_{23^{2}}$ is 1.529.m 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-493}) \)$)$ |
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.a_am | $4$ | (not in LMFDB) |