Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 8 x + 32 x^{2} - 88 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.0750991438595$, $\pm0.424900856141$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{6})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $3$ |
| Cyclic group of points: | yes |
This isogeny class is simple but not geometrically 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $58$ | $14500$ | $1760938$ | $210250000$ | $25769191258$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $122$ | $1324$ | $14358$ | $160004$ | $1771562$ | $19496684$ | $214377118$ | $2357916004$ | $25937424602$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):
- $y^2=2 x^6+2 x^5+9 x^4+9 x^3+10 x^2+6 x+8$
- $y^2=6 x^6+8 x^5+3 x^4+x^3+3 x+10$
- $y^2=10 x^6+3 x^5+2 x^4+2 x^2+8 x+10$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11^{4}}$.
Endomorphism algebra over $\F_{11}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{6})\). |
| The base change of $A$ to $\F_{11^{4}}$ is 1.14641.afm 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$ |
- Endomorphism algebra over $\F_{11^{2}}$
The base change of $A$ to $\F_{11^{2}}$ is the simple isogeny class 2.121.a_afm and its endomorphism algebra is \(\Q(i, \sqrt{6})\).
Base change
This is a primitive isogeny class.