Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 4 x + 8 x^{2} - 44 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.109781100996$, $\pm0.609781100996$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\zeta_{8})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $8$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $82$ | $14596$ | $1643362$ | $213043216$ | $26085367762$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $122$ | $1232$ | $14550$ | $161968$ | $1771562$ | $19487896$ | $214413214$ | $2358048392$ | $25937424602$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=10 x^6+x^5+8 x^4+4 x^3+8 x^2+7 x+7$
- $y^2=6 x^6+6 x^5+9 x^4+7 x^3+x$
- $y^2=3 x^6+7 x^5+5 x^4+2 x^3+2 x+5$
- $y^2=2 x^6+7 x^5+3 x^4+9 x^3+x+7$
- $y^2=5 x^6+x^5+6 x^4+3 x^3+4 x^2+4 x+10$
- $y^2=8 x^5+9 x^4+9 x^3+x+6$
- $y^2=10 x^6+2 x^5+2 x^4+2 x^2+9 x+10$
- $y^2=9 x^5+8 x^3+2 x^2+5$
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 \(\Q(\zeta_{8})\). |
| The base change of $A$ to $\F_{11^{4}}$ is 1.14641.abu 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$ |
- Endomorphism algebra over $\F_{11^{2}}$
The base change of $A$ to $\F_{11^{2}}$ is the simple isogeny class 2.121.a_abu and its endomorphism algebra is \(\Q(\zeta_{8})\).
Base change
This is a primitive isogeny class.