Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 8 x + 32 x^{2} + 184 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.450780744796$, $\pm0.950780744796$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{30})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $10$ |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $754$ | $278980$ | $151681426$ | $77829840400$ | $41427077615794$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $32$ | $530$ | $12464$ | $278118$ | $6436432$ | $148035890$ | $3404983264$ | $78310618558$ | $1801150498592$ | $41426511213650$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=20 x^6+4 x^5+x^4+11 x^3+3 x^2+12 x+19$
- $y^2=6 x^6+16 x^5+14 x^4+18 x^3+6 x+12$
- $y^2=x^6+11 x^5+12 x^4+2 x^3+18 x^2+3 x+2$
- $y^2=2 x^6+14 x^5+17 x^4+12 x^3+7 x^2+21 x+6$
- $y^2=21 x^6+17 x^5+x^4+14 x^3+11 x^2+15 x+3$
- $y^2=12 x^6+11 x^5+17 x^4+17 x^2+12 x+12$
- $y^2=14 x^6+10 x^5+7 x^4+7 x^2+13 x+14$
- $y^2=18 x^6+20 x^5+7 x^4+8 x^3+11 x^2+22 x+8$
- $y^2=16 x^6+2 x^4+x^3+20 x^2+5 x+15$
- $y^2=16 x^6+9 x^5+12 x^4+20 x^2+4 x+1$
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 \(\Q(i, \sqrt{30})\). |
| The base change of $A$ to $\F_{23^{4}}$ is 1.279841.abhe 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-30}) \)$)$ |
- Endomorphism algebra over $\F_{23^{2}}$
The base change of $A$ to $\F_{23^{2}}$ is the simple isogeny class 2.529.a_abhe and its endomorphism algebra is \(\Q(i, \sqrt{30})\).
Base change
This is a primitive isogeny class.