Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 6 x + 13 x^{2} - 138 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.0484559505595$, $\pm0.618210716107$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $399$ | $274113$ | $143280900$ | $78043534569$ | $41434191197679$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $520$ | $11772$ | $278884$ | $6437538$ | $148006150$ | $3404708766$ | $78311343364$ | $1801151591076$ | $41426499766600$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):
- $y^2=7 x^6+7 x^5+6 x^4+12 x^3+19 x^2+4 x+12$
- $y^2=3 x^6+x^5+22 x^4+20 x^3+18 x^2+19 x+14$
- $y^2=5 x^6+6 x^5+7 x^4+11 x^3+6 x^2+x+5$
- $y^2=20 x^6+20 x^4+15 x^3+21 x^2+20 x+13$
- $y^2=14 x^6+2 x^5+5 x^4+3 x^3+12 x^2+14 x+21$
- $y^2=19 x^6+16 x^5+20 x^4+14 x^3+12 x^2+5 x+9$
- $y^2=7 x^6+6 x^5+2 x^4+17 x^3+12 x^2+3 x+18$
- $y^2=3 x^6+16 x^5+4 x^4+3 x^3+7 x^2+21 x+8$
- $y^2=15 x^6+16 x^5+14 x^4+x^3+6 x^2+12 x+14$
- $y^2=2 x^6+18 x^5+20 x^4+4 x^3+20 x^2+18 x+14$
- $y^2=21 x^6+8 x^5+10 x^4+10 x^3+12 x^2+18 x+21$
- $y^2=15 x^6+5 x^5+12 x^4+15 x^3+15 x^2+17$
- $y^2=12 x^6+10 x^4+8 x^3+20 x^2+9 x+5$
- $y^2=19 x^6+11 x^5+7 x^4+2 x^3+5 x^2+8 x+12$
- $y^2=19 x^6+2 x^5+17 x^4+7 x^3+16 x^2+20 x+19$
- $y^2=4 x^6+16 x^5+x^4+8 x^3+4 x^2+8 x+4$
- $y^2=9 x^6+21 x^5+x^4+12 x^3+2 x^2+5 x+9$
- $y^2=5 x^6+3 x^5+12 x^4+7 x^2+16 x+21$
- $y^2=11 x^6+15 x^5+20 x^4+6 x^3+18 x^2+5 x+11$
- $y^2=15 x^6+13 x^5+17 x^4+8 x^3+16 x^2+20 x+21$
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.ahq 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-14}) \)$)$ |
Base change
This is a primitive isogeny class.