Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 6 x + 23 x^{2} )^{2}$ |
| $1 + 12 x + 82 x^{2} + 276 x^{3} + 529 x^{4}$ | |
| Frobenius angles: | $\pm0.715122617226$, $\pm0.715122617226$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $14$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 3, 5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $900$ | $291600$ | $143280900$ | $78848640000$ | $41411155522500$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $36$ | $550$ | $11772$ | $281758$ | $6433956$ | $148006150$ | $3405058812$ | $78310269118$ | $1801151591076$ | $41426534107750$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):
- $y^2=16 x^6+5 x^4+8 x^3+5 x^2+16$
- $y^2=9 x^6+17 x^5+17 x^4+17 x^2+17 x+9$
- $y^2=9 x^6+4 x^4+6 x^3+4 x^2+9$
- $y^2=18 x^6+10 x^5+8 x^4+15 x^3+18 x^2+19 x+16$
- $y^2=17 x^6+15 x^4+15 x^2+17$
- $y^2=3 x^6+2 x^4+2 x^2+3$
- $y^2=9 x^6+6 x^4+9 x^3+6 x^2+9$
- $y^2=14 x^6+6 x^5+7 x^4+7 x^3+15 x^2+13 x+19$
- $y^2=17 x^5+19 x^4+10 x^3+19 x^2+17 x$
- $y^2=x^6+17 x^5+11 x^4+17 x^3+11 x^2+17 x+1$
- $y^2=22 x^6+7 x^5+x^4+20 x^3+2 x^2+5 x+15$
- $y^2=6 x^6+22 x^5+14 x^4+10 x^3+11 x^2+11 x+4$
- $y^2=11 x^6+8 x^5+15 x^4+9 x^3+20 x^2+4 x+15$
- $y^2=2 x^6+16 x^5+9 x^4+x^2+9 x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23}$.
Endomorphism algebra over $\F_{23}$| The isogeny class factors as 1.23.g 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-14}) \)$)$ |
Base change
This is a primitive isogeny class.