Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 6 x + 28 x^{2} - 66 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.247161792509$, $\pm0.438778135579$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.131904.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $10$ |
| Isomorphism classes: | 10 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $78$ | $17316$ | $1896102$ | $215341776$ | $25926727398$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $142$ | $1422$ | $14710$ | $160986$ | $1772782$ | $19489938$ | $214328734$ | $2357777862$ | $25937284702$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=8 x^6+5 x^5+x^4+x^3+10 x^2+2 x+8$
- $y^2=8 x^6+x^5+6 x^4+3 x^3+7 x^2+4 x$
- $y^2=4 x^6+x^4+6 x^3+10 x^2+4 x+3$
- $y^2=4 x^6+4 x^5+9 x^4+5 x^3+10 x^2+3 x+8$
- $y^2=3 x^6+2 x^5+5 x^4+7 x^3+5 x^2+9 x+7$
- $y^2=2 x^6+5 x^5+3 x^4+8 x^3+3 x^2+9 x+10$
- $y^2=7 x^6+7 x^5+7 x^4+x^3+5 x^2+10$
- $y^2=5 x^6+8 x^5+8 x^4+8 x+6$
- $y^2=7 x^6+3 x^4+9 x^3+9 x^2+3 x+8$
- $y^2=9 x^6+3 x^5+8 x^4+5 x^3+10 x^2+7 x+7$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11}$.
Endomorphism algebra over $\F_{11}$| The endomorphism algebra of this simple isogeny class is 4.0.131904.1. |
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 |
|---|---|---|
| 2.11.g_bc | $2$ | 2.121.u_ja |