Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - x + 11 x^{2} - 11 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.302650775488$, $\pm0.641582372415$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-130 -6 \sqrt{5}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $12$ |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $1$ |
| Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $121$ | $17545$ | $1767931$ | $217926445$ | $26035000816$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $11$ | $143$ | $1331$ | $14883$ | $161656$ | $1766963$ | $19478701$ | $214378483$ | $2357936801$ | $25937661398$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=2 x^6+5 x^5+8 x^4+2 x^3+x^2+8 x$
- $y^2=3 x^6+3 x^5+4 x^4+6 x^3+8 x^2+4 x+6$
- $y^2=5 x^6+6 x^5+6 x^4+4 x^2+5 x+6$
- $y^2=x^6+7 x^5+9 x^4+8 x^3+x+10$
- $y^2=5 x^5+2 x^4+8 x^3+9 x^2+1$
- $y^2=5 x^6+7 x^4+5 x^3+3 x^2+10 x+6$
- $y^2=7 x^6+x^5+9 x^4+4 x^3+10 x^2+5 x+9$
- $y^2=6 x^5+6 x^4+9 x^3+x^2+x+3$
- $y^2=7 x^6+3 x^5+9 x^4+8 x^3+4 x^2+9 x+2$
- $y^2=7 x^6+5 x^5+6 x^4+2 x^3+2 x^2+8 x+2$
- $y^2=10 x^6+5 x^5+9 x^3+6 x^2+6 x$
- $y^2=2 x^5+7 x^3+8 x^2+6 x+9$
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 \(\Q(\sqrt{-130 -6 \sqrt{5}})\). |
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_l | $2$ | 2.121.v_nd |