Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - x + 23 x^{2} )^{2}$ |
| $1 - 2 x + 47 x^{2} - 46 x^{3} + 529 x^{4}$ | |
| Frobenius angles: | $\pm0.466753484570$, $\pm0.466753484570$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $9$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $529$ | $330625$ | $149719696$ | $77771265625$ | $41393949718969$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $22$ | $620$ | $12304$ | $277908$ | $6431282$ | $148075310$ | $3404981294$ | $78310234468$ | $1801148326192$ | $41426524147100$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=8 x^6+9 x^5+12 x^4+8 x^3+7 x+1$
- $y^2=11 x^6+6 x^5+2 x^4+6 x^3+2 x^2+6 x+11$
- $y^2=3 x^6+22 x^5+16 x^4+15 x^3+16 x^2+22 x+3$
- $y^2=20 x^6+14 x^5+22 x^3+14 x+20$
- $y^2=22 x^6+21 x^5+19 x^4+13 x^3+3 x^2+21 x+10$
- $y^2=x^6+12 x^5+19 x^4+14 x^3+17 x^2+8 x+22$
- $y^2=7 x^6+12 x^5+9 x^4+18 x^3+17 x^2+7 x+13$
- $y^2=7 x^6+17 x^5+22 x^4+21 x^3+22 x^2+17 x+7$
- $y^2=19 x^6+9 x^5+8 x^4+22 x^3+8 x^2+9 x+19$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23}$.
Endomorphism algebra over $\F_{23}$| The isogeny class factors as 1.23.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-91}) \)$)$ |
Base change
This is a primitive isogeny class.