Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 4 x + 23 x^{2} )^{2}$ |
| $1 + 8 x + 62 x^{2} + 184 x^{3} + 529 x^{4}$ | |
| Frobenius angles: | $\pm0.636928592136$, $\pm0.636928592136$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $14$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 7$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $784$ | $313600$ | $142945936$ | $78400000000$ | $41481173785744$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $32$ | $590$ | $11744$ | $280158$ | $6444832$ | $147994670$ | $3404795104$ | $78312054718$ | $1801149081632$ | $41426500935950$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):
- $y^2=x^6+6 x^5+22 x^4+21 x^3+22 x^2+6 x+1$
- $y^2=22 x^6+22 x^4+22 x^2+22$
- $y^2=4 x^6+21 x^5+22 x^4+21 x^3+22 x^2+20$
- $y^2=x^6+21 x^5+7 x^4+10 x^2+10 x+6$
- $y^2=7 x^5+18 x^4+7 x^3+3 x^2+12 x+17$
- $y^2=x^6+9 x^5+17 x^4+22 x^3+20 x^2+8 x+3$
- $y^2=16 x^6+18 x^5+17 x^4+4 x^3+17 x^2+18 x+16$
- $y^2=7 x^5+3 x^4+8 x^3+18 x^2+2 x+2$
- $y^2=4 x^6+20 x^4+20 x^2+4$
- $y^2=18 x^6+14 x^5+17 x^4+12 x^3+17 x^2+14 x+18$
- $y^2=21 x^6+21 x^5+2 x^4+21 x^3+5 x^2+x$
- $y^2=8 x^6+16 x^4+8 x^3+6 x^2+13$
- $y^2=9 x^6+12 x^5+x^4+7 x^3+x^2+12 x+9$
- $y^2=13 x^6+12 x^4+12 x^2+13$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23}$.
Endomorphism algebra over $\F_{23}$| The isogeny class factors as 1.23.e 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$ |
Base change
This is a primitive isogeny class.