Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 14 x + 90 x^{2} - 322 x^{3} + 529 x^{4}$ |
| Frobenius angles: | $\pm0.0869454733845$, $\pm0.334554373298$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.11600.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
This isogeny class is simple and 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $284$ | $271504$ | $148878764$ | $78319129856$ | $41403008393164$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $514$ | $12238$ | $279870$ | $6432690$ | $148013218$ | $3404808886$ | $78311602494$ | $1801157425594$ | $41426529022914$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=7 x^6+7 x^5+19 x^4+4 x^3+5 x^2+16 x+20$
- $y^2=15 x^6+6 x^4+19 x^3+9 x^2+11 x+14$
- $y^2=21 x^6+17 x^5+20 x^4+11 x^2+14 x+19$
- $y^2=11 x^5+7 x^4+12 x^3+16 x^2+11 x+5$
- $y^2=10 x^6+16 x^5+7 x^4+21 x^3+8 x^2+9 x+5$
- $y^2=19 x^6+13 x^5+8 x^4+3 x^3+7 x^2+6 x+10$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23}$.
Endomorphism algebra over $\F_{23}$| The endomorphism algebra of this simple isogeny class is 4.0.11600.1. |
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.o_dm | $2$ | (not in LMFDB) |