Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 6 x + 19 x^{2} - 66 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.0720381525652$, $\pm0.522289064218$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.13968.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $69$ | $14697$ | $1679184$ | $208829673$ | $25927023309$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $124$ | $1260$ | $14260$ | $160986$ | $1773430$ | $19483134$ | $214341028$ | $2358048564$ | $25937876524$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=2 x^6+2 x^5+x^4+3 x^3+x^2+6 x+2$
- $y^2=6 x^6+10 x^5+10 x^4+6 x^3+3 x^2+6 x+7$
- $y^2=5 x^6+8 x^5+2 x^4+9 x^3+7 x^2+2 x+2$
- $y^2=9 x^6+10 x^5+2 x^4+6 x^2+8 x+8$
- $y^2=8 x^6+4 x^5+7 x^4+10 x^3+x^2+10$
- $y^2=2 x^6+2 x^5+9 x^4+6 x^3+9 x^2+10 x+8$
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 4.0.13968.2. |
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_t | $2$ | 2.121.c_ahh |