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: | $\pm0.533246515430$, $\pm0.533246515430$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $625$ | $330625$ | $146410000$ | $77771265625$ | $41459111265625$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $26$ | $620$ | $12032$ | $277908$ | $6441406$ | $148075310$ | $3404669602$ | $78310234468$ | $1801156996736$ | $41426524147100$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=17 x^6+22 x^5+14 x^4+17 x^3+12 x+5$
- $y^2=16 x^6+15 x^5+5 x^4+15 x^3+5 x^2+15 x+16$
- $y^2=19 x^6+9 x^5+17 x^4+3 x^3+17 x^2+9 x+19$
- $y^2=4 x^6+12 x^5+9 x^3+12 x+4$
- $y^2=9 x^6+18 x^5+13 x^4+21 x^3+19 x^2+18 x+2$
- $y^2=5 x^6+14 x^5+3 x^4+x^3+16 x^2+17 x+18$
- $y^2=6 x^6+7 x^5+11 x^4+22 x^3+8 x^2+6 x+21$
- $y^2=6 x^6+8 x^5+9 x^4+18 x^3+9 x^2+8 x+6$
- $y^2=3 x^6+22 x^5+17 x^4+18 x^3+17 x^2+22 x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23}$.
Endomorphism algebra over $\F_{23}$| The isogeny class factors as 1.23.b 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-91}) \)$)$ |
Base change
This is a primitive isogeny class.