Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 11 x + 93 x^{2} - 477 x^{3} + 2139 x^{4} - 5819 x^{5} + 12167 x^{6}$ |
| Frobenius angles: | $\pm0.242559439977$, $\pm0.328869131452$, $\pm0.528273725691$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\zeta_{7})\) |
| Galois group: | $C_6$ |
| 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: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $8093$ | $167581751$ | $1847179396091$ | $21964774030054759$ | $266798506515902581403$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $13$ | $595$ | $12475$ | $280483$ | $6440283$ | $148036003$ | $3404621520$ | $78310234947$ | $1801154996173$ | $41426524283235$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 13 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=22 x^7+10 x^5+7 x^4+17 x^3+4 x^2+6 x+22$
- $y^2=22 x^7+11 x^5+x^4+5 x^3+8 x^2+19 x+17$
- $y^2=22 x^7+22 x^5+5 x^4+3 x^3+21 x^2+21 x+10$
- $y^2=x^7+22 x^5+10 x^3+3 x^2+14 x+15$
- $y^2=22 x^7+19 x^5+17 x^4+13 x^3+x^2+12 x+17$
- $y^2=x^7+18 x^5+17 x^3+10 x^2+5 x+14$
- $y^2=22 x^7+5 x^5+14 x^4+21 x^3+11 x^2+6 x+20$
- $y^2=x^7+9 x^5+22 x^4+4 x^3+13 x^2+22 x+15$
- $y^2=22 x^7+22 x^5+18 x^4+2 x^3+22 x^2+16 x+10$
- $y^2=x^7+10 x^5+18 x^4+21 x^3+16 x+17$
- $y^2=x^7+3 x^5+10 x^4+14 x^3+15 x^2+10 x+7$
- $y^2=x^7+7 x^5+9 x^4+17 x^2+22 x+11$
- $y^2=x^7+13 x^5+9 x^4+2 x^3+16 x^2+11 x+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23^{7}}$.
Endomorphism algebra over $\F_{23}$| The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{7})\). |
| The base change of $A$ to $\F_{23^{7}}$ is 1.3404825447.adwom 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$ |
Base change
This is a primitive isogeny class.