Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 2 x + 23 x^{2} )^{2}$ |
| $1 - 4 x + 50 x^{2} - 92 x^{3} + 529 x^{4}$ | |
| Frobenius angles: | $\pm0.433137181604$, $\pm0.433137181604$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $8$ |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $484$ | $327184$ | $151240804$ | $77916906496$ | $41369877891364$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $614$ | $12428$ | $278430$ | $6427540$ | $148050758$ | $3405057676$ | $78311107774$ | $1801147565204$ | $41426498203814$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=17 x^6+2 x^5+11 x^4+14 x^3+10 x^2+18 x+22$
- $y^2=14 x^6+9 x^5+16 x^4+4 x^3+12 x^2+14 x+15$
- $y^2=18 x^6+6 x^5+7 x^4+4 x^3+2 x^2+20 x+8$
- $y^2=5 x^6+18 x^5+6 x^4+16 x^3+6 x^2+18 x+5$
- $y^2=7 x^6+19 x^4+19 x^2+7$
- $y^2=12 x^6+17 x^5+3 x^4+12 x^3+22 x^2+16 x+5$
- $y^2=15 x^6+15 x^5+16 x^4+13 x^3+5 x^2+5 x+15$
- $y^2=20 x^6+2 x^5+18 x^4+22 x^3+11 x^2+16 x+8$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23}$.
Endomorphism algebra over $\F_{23}$| The isogeny class factors as 1.23.ac 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-22}) \)$)$ |
Base change
This is a primitive isogeny class.