Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 5 x + 11 x^{2} )( 1 + 5 x + 11 x^{2} )$ |
| $1 - 3 x^{2} + 121 x^{4}$ | |
| Frobenius angles: | $\pm0.228229222880$, $\pm0.771770777120$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $8$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $119$ | $14161$ | $1772624$ | $221265625$ | $25937221079$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $116$ | $1332$ | $15108$ | $161052$ | $1773686$ | $19487172$ | $214308868$ | $2357947692$ | $25937017556$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=x^6+8 x^5+4 x^4+8 x^3+6 x^2+7 x+2$
- $y^2=10 x^6+8 x^5+4 x^4+3 x^3+9 x^2+8$
- $y^2=9 x^6+5 x^5+8 x^4+6 x^3+7 x^2+5$
- $y^2=4 x^6+3 x^5+7 x^4+7 x^3+x^2+5 x+2$
- $y^2=10 x^6+10 x^4+9 x^3+10 x^2+10$
- $y^2=9 x^6+9 x^4+7 x^3+9 x^2+9$
- $y^2=7 x^6+x^5+x^4+9 x^3+7 x^2+x+1$
- $y^2=3 x^6+2 x^5+2 x^4+7 x^3+3 x^2+2 x+2$
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.af $\times$ 1.11.f 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.ad 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$ |
Base change
This is a primitive isogeny class.