Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 6 x + 25 x^{2} - 66 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.193114434330$, $\pm0.473552232337$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $11$ |
| Isomorphism classes: | 7 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $75$ | $16425$ | $1822500$ | $213672825$ | $26013826875$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $136$ | $1368$ | $14596$ | $161526$ | $1776238$ | $19495986$ | $214331716$ | $2357815608$ | $25937327176$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 11 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=10x^6+8x^5+4x^4+x^3+6x^2+6x+5$
- $y^2=3x^6+6x^5+8x^4+5x^3+5x^2+10x+4$
- $y^2=2x^6+2x^5+3x^3+7x^2+8x+2$
- $y^2=10x^6+3x^5+8x^4+5x^3+7x^2+3x+2$
- $y^2=6x^6+9x^5+8x^4+x^3+5x^2+10x+3$
- $y^2=8x^6+4x^4+2x^3+8x^2+2x+5$
- $y^2=6x^6+2x^5+x^4+2x^3+10x^2+8x+10$
- $y^2=10x^6+6x^5+4x^4+7x^2+2$
- $y^2=8x^6+2x^5+3x^4+3x^3+9x^2+x+10$
- $y^2=2x^6+8x^5+5x^4+2x^3+5x^2+7x+10$
- $y^2=2x^6+4x^5+4x^2+7x+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11^{3}}$.
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 $\F_{11^{3}}$ is 1.1331.s 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$ |
Base change
This is a primitive isogeny class.