Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 6 x + 11 x^{2} )( 1 + 6 x + 11 x^{2} )$ |
| $1 - 14 x^{2} + 121 x^{4}$ | |
| Frobenius angles: | $\pm0.140218899004$, $\pm0.859781100996$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $6$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 3$ |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $108$ | $11664$ | $1773900$ | $215737344$ | $25937522028$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $94$ | $1332$ | $14734$ | $161052$ | $1776238$ | $19487172$ | $214413214$ | $2357947692$ | $25937619454$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=x^6+6 x^5+2 x^3+7 x+7$
- $y^2=9 x^6+9 x^5+9 x^4+6 x^3+2 x^2+9 x+2$
- $y^2=x^5+x$
- $y^2=2 x^5+10 x^4+8 x^3+4 x^2+10 x$
- $y^2=9 x^6+x^5+9 x^4+4 x^3+6 x^2+9 x+10$
- $y^2=6 x^6+x^5+x^4+8 x^3+10 x^2+x+5$
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.ag $\times$ 1.11.g and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
| The base change of $A$ to $\F_{11^{2}}$ is 1.121.ao 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$ |
Base change
This is a primitive isogeny class.