Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 24 x^{2} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.162641718363$, $\pm0.837358281637$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-22}, \sqrt{70})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $4$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $506$ | $256036$ | $148060154$ | $78581544976$ | $41426506234586$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $24$ | $482$ | $12168$ | $280806$ | $6436344$ | $148084418$ | $3404825448$ | $78311639998$ | $1801152661464$ | $41426501255522$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=20 x^6+14 x^5+5 x^4+17 x^3+17 x^2+18 x$
- $y^2=8 x^6+x^5+2 x^4+16 x^3+16 x^2+21 x$
- $y^2=20 x^6+13 x^5+3 x^4+x^3+2 x^2+2 x+14$
- $y^2=8 x^6+19 x^5+15 x^4+5 x^3+10 x^2+10 x+1$
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 \(\Q(\sqrt{-22}, \sqrt{70})\). |
| The base change of $A$ to $\F_{23^{2}}$ is 1.529.ay 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-385}) \)$)$ |
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_y | $4$ | (not in LMFDB) |