Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 9 x^{2} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.218658821422$, $\pm0.781341178578$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-37}, \sqrt{55})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $4$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $521$ | $271441$ | $148049444$ | $78859310761$ | $41426500489961$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $24$ | $512$ | $12168$ | $281796$ | $6436344$ | $148062998$ | $3404825448$ | $78310195588$ | $1801152661464$ | $41426489766272$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=19 x^6+x^5+18 x^3+19 x^2+2$
- $y^2=15 x^6+4 x^5+3 x^4+12 x^3+18 x^2+11 x+12$
- $y^2=4 x^6+22 x^5+15 x^4+14 x^3+22 x^2+9$
- $y^2=9 x^6+11 x^5+9 x^4+20 x^3+2 x^2+8 x+8$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23^{2}}$.
Endomorphism algebra over $\F_{23}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-37}, \sqrt{55})\). |
| The base change of $A$ to $\F_{23^{2}}$ is 1.529.aj 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2035}) \)$)$ |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.23.a_j | $4$ | (not in LMFDB) |