Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 11 x^{2} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.333333333333$, $\pm0.666666666667$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{11})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $9$ |
| Cyclic group of points: | yes |
This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $133$ | $17689$ | $1768900$ | $217946169$ | $25937585653$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $144$ | $1332$ | $14884$ | $161052$ | $1766238$ | $19487172$ | $214388164$ | $2357947692$ | $25937746704$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=10 x^6+2 x^5+x^4+3 x^3+8 x^2+6$
- $y^2=9 x^6+4 x^5+2 x^4+6 x^3+5 x^2+1$
- $y^2=4 x^6+4 x^5+4 x^4+7 x^3+3 x^2+4 x+10$
- $y^2=8 x^6+8 x^5+8 x^4+3 x^3+6 x^2+8 x+9$
- $y^2=8 x^6+2 x^5+7 x^4+2 x^2+7 x+10$
- $y^2=5 x^6+4 x^5+3 x^4+4 x^2+3 x+9$
- $y^2=3 x^6+4 x^5+2 x^4+7 x^2+7$
- $y^2=6 x^6+8 x^5+4 x^4+3 x^2+3$
- $y^2=x^6+2 x^5+x^4+3 x^3+7 x^2+5 x+9$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11^{6}}$.
Endomorphism algebra over $\F_{11}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{11})\). |
| The base change of $A$ to $\F_{11^{6}}$ is 1.1771561.adyk 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $11$ and $\infty$. |
- Endomorphism algebra over $\F_{11^{2}}$
The base change of $A$ to $\F_{11^{2}}$ is 1.121.l 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ - Endomorphism algebra over $\F_{11^{3}}$
The base change of $A$ to $\F_{11^{3}}$ is the simple isogeny class 2.1331.a_adyk and its endomorphism algebra is the quaternion algebra over \(\Q(\sqrt{11}) \) ramified at both real infinite places.
Base change
This is a primitive isogeny class.