Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - x + 19 x^{2} - 11 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.387147268865$, $\pm0.562925372531$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.276653.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $129$ | $19737$ | $1802259$ | $211363533$ | $25918537584$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $11$ | $159$ | $1355$ | $14435$ | $160936$ | $1771155$ | $19483909$ | $214387123$ | $2358047489$ | $25937001174$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):
- $y^2=2 x^6+x^5+6 x^4+5 x^3+5 x^2+2 x+8$
- $y^2=8 x^6+7 x^5+7 x^4+8 x^3+8 x^2+x+8$
- $y^2=9 x^6+10 x^5+3 x^4+x^3+4 x^2+2 x+8$
- $y^2=3 x^6+8 x^5+2 x^4+2 x^3+5 x^2+x+3$
- $y^2=10 x^6+6 x^5+10 x^4+8 x^3+2 x^2+9 x+10$
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 4.0.276653.1. |
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.b_t | $2$ | 2.121.bl_wj |