Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 11 x^{2} )^{2}$ |
| $1 + 22 x^{2} + 121 x^{4}$ | |
| Frobenius angles: | $\pm0.5$, $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Jacobians: | $8$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 3$ |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $144$ | $20736$ | $1774224$ | $207360000$ | $25937746704$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $166$ | $1332$ | $14158$ | $161052$ | $1776886$ | $19487172$ | $214300318$ | $2357947692$ | $25938068806$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=x^6+x^3+5$
- $y^2=2 x^6+2 x^3+10$
- $y^2=10 x^6+3 x^5+10 x^4+4 x^3+7 x^2+4 x+2$
- $y^2=9 x^6+6 x^5+9 x^4+8 x^3+3 x^2+8 x+4$
- $y^2=x^6+10$
- $y^2=x^6+3$
- $y^2=2 x^6+6$
- $y^2=x^5+9 x^3+3 x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11^{2}}$.
Endomorphism algebra over $\F_{11}$| The isogeny class factors as 1.11.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$ |
| The base change of $A$ to $\F_{11^{2}}$ is 1.121.w 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $11$ and $\infty$. |
Base change
This is a primitive isogeny class.