Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 10 x^{2} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.174900856141$, $\pm0.825099143859$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $13$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $112$ | $12544$ | $1774192$ | $218566656$ | $25937197552$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $102$ | $1332$ | $14926$ | $161052$ | $1776822$ | $19487172$ | $214377118$ | $2357947692$ | $25936970502$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 13 curves (of which all are hyperelliptic):
- $y^2=5 x^5+x^4+9 x^3+2 x^2+4 x+1$
- $y^2=7 x^6+x^5+5 x^4+3 x^3+8 x^2+3 x+5$
- $y^2=x^6+7 x^5+5 x^4+6 x^2+x+10$
- $y^2=9 x^6+4 x^5+9 x^4+9 x^3+8 x^2+9 x+7$
- $y^2=9 x^6+8 x^5+4 x^4+6 x^2+7 x+7$
- $y^2=3 x^5+x^4+7 x^3+8 x^2+3$
- $y^2=2 x^6+3 x^5+9 x^4+5 x^3+x^2+2 x+5$
- $y^2=2 x^6+2 x^5+10 x^4+6 x^3+x^2+2 x+9$
- $y^2=7 x^6+2 x^5+10 x^4+9 x^3+9 x^2+8 x+1$
- $y^2=9 x^5+3 x^4+2 x^3+4 x^2+9 x+6$
- $y^2=2 x^6+9 x^5+7 x^4+3 x^2+8 x+5$
- $y^2=x^6+x^5+7 x^4+7 x^3+x^2+9 x+6$
- $y^2=x^6+3 x^4+7 x^3+4 x^2+10 x+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11^{2}}$.
Endomorphism algebra over $\F_{11}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{2}, \sqrt{-3})\). |
| The base change of $A$ to $\F_{11^{2}}$ is 1.121.ak 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$ |
Base change
This is a primitive isogeny class.