Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 6 x + 13 x^{2} + 138 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.381789283893$, $\pm0.951544049441$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-14})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $20$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $687$ | $274113$ | $152917956$ | $78043534569$ | $41418821208207$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $30$ | $520$ | $12564$ | $278884$ | $6435150$ | $148006150$ | $3404942130$ | $78311343364$ | $1801153731852$ | $41426499766600$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):
- $y^2=6 x^6+6 x^5+15 x^4+7 x^3+13 x^2+10 x+7$
- $y^2=15 x^6+5 x^5+18 x^4+8 x^3+21 x^2+3 x+1$
- $y^2=2 x^6+7 x^5+12 x^4+9 x^3+7 x^2+5 x+2$
- $y^2=4 x^6+4 x^4+3 x^3+18 x^2+4 x+21$
- $y^2=12 x^6+5 x^5+x^4+19 x^3+7 x^2+12 x+18$
- $y^2=3 x^6+11 x^5+8 x^4+x^3+14 x^2+2 x+22$
- $y^2=12 x^6+7 x^5+10 x^4+16 x^3+14 x^2+15 x+21$
- $y^2=15 x^6+11 x^5+20 x^4+15 x^3+12 x^2+13 x+17$
- $y^2=3 x^6+17 x^5+12 x^4+14 x^3+15 x^2+7 x+12$
- $y^2=10 x^6+21 x^5+8 x^4+20 x^3+8 x^2+21 x+1$
- $y^2=18 x^6+20 x^5+2 x^4+2 x^3+7 x^2+22 x+18$
- $y^2=6 x^6+2 x^5+14 x^4+6 x^3+6 x^2+16$
- $y^2=14 x^6+4 x^4+17 x^3+8 x^2+22 x+2$
- $y^2=13 x^6+16 x^5+6 x^4+5 x^3+x^2+20 x+7$
- $y^2=3 x^6+10 x^5+16 x^4+12 x^3+11 x^2+8 x+3$
- $y^2=20 x^6+11 x^5+5 x^4+17 x^3+20 x^2+17 x+20$
- $y^2=22 x^6+13 x^5+5 x^4+14 x^3+10 x^2+2 x+22$
- $y^2=x^6+19 x^5+7 x^4+6 x^2+17 x+18$
- $y^2=9 x^6+6 x^5+8 x^4+7 x^3+21 x^2+2 x+9$
- $y^2=6 x^6+19 x^5+16 x^4+17 x^3+11 x^2+8 x+13$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23^{3}}$.
Endomorphism algebra over $\F_{23}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-14})\). |
| The base change of $A$ to $\F_{23^{3}}$ is 1.12167.hq 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-14}) \)$)$ |
Base change
This is a primitive isogeny class.