Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 27 x^{2} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.150162972524$, $\pm0.849837027476$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-19}, \sqrt{73})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $503$ | $253009$ | $148059056$ | $78495789241$ | $41426511147743$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $24$ | $476$ | $12168$ | $280500$ | $6436344$ | $148082222$ | $3404825448$ | $78311888164$ | $1801152661464$ | $41426511081836$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=9 x^6+6 x^5+12 x^4+20 x^3+19 x^2+6$
- $y^2=22 x^6+7 x^5+14 x^4+8 x^3+3 x^2+7$
- $y^2=8 x^6+4 x^5+19 x^4+5 x^3+3 x^2+8 x+11$
- $y^2=15 x^6+13 x^5+18 x^4+15 x^3+22 x^2+18 x+13$
- $y^2=20 x^6+13 x^5+7 x^4+21 x^3+19 x+13$
- $y^2=8 x^6+19 x^5+12 x^4+13 x^3+3 x+19$
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 \(\Q(\sqrt{-19}, \sqrt{73})\). |
| The base change of $A$ to $\F_{23^{2}}$ is 1.529.abb 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1387}) \)$)$ |
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_bb | $4$ | (not in LMFDB) |