Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 10 x + 50 x^{2} - 230 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.0138570610432$, $\pm0.486142938957$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{21})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $4$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is simple but not geometrically 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $340$ | $278800$ | $145742020$ | $77729440000$ | $41387589633700$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $14$ | $530$ | $11978$ | $277758$ | $6430294$ | $148035890$ | $3404749138$ | $78309933118$ | $1801149155774$ | $41426511213650$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=18 x^6+16 x^5+4 x^4+19 x^2+2 x+20$
- $y^2=20 x^6+4 x^5+19 x+20$
- $y^2=10 x^6+3 x^5+4 x^4+4 x^2+20 x+10$
- $y^2=20 x^6+6 x^5+19 x^4+18 x^3+14 x^2+22 x+15$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23^{4}}$.
Endomorphism algebra over $\F_{23}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{21})\). |
| The base change of $A$ to $\F_{23^{4}}$ is 1.279841.aboc 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-21}) \)$)$ |
- Endomorphism algebra over $\F_{23^{2}}$
The base change of $A$ to $\F_{23^{2}}$ is the simple isogeny class 2.529.a_aboc and its endomorphism algebra is \(\Q(i, \sqrt{21})\).
Base change
This is a primitive isogeny class.