Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 6 x + 11 x^{2} )( 1 + 11 x^{2} )$ |
$1 - 6 x + 22 x^{2} - 66 x^{3} + 121 x^{4}$ | |
Frobenius angles: | $\pm0.140218899004$, $\pm0.5$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $16$ |
Isomorphism classes: | 80 |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $72$ | $15552$ | $1750248$ | $211507200$ | $26014085352$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $130$ | $1314$ | $14446$ | $161526$ | $1776562$ | $19495986$ | $214356766$ | $2358013734$ | $25937844130$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=7x^5+8x^4+2x^3+2x^2+9x+1$
- $y^2=2x^6+6x^5+4x^4+5x^3+4x^2+6x+2$
- $y^2=8x^6+3x^5+3x^4+4x^3+3x^2+3x+8$
- $y^2=2x^6+8x^5+4x^4+2x^3+x^2+6x+10$
- $y^2=6x^6+2x^5+5x^4+4x^3+5x^2+2x+6$
- $y^2=8x^6+2x^5+x^4+3x^2+10x+2$
- $y^2=3x^5+9x^4+7x^3+8x^2+8x+8$
- $y^2=10x^5+10x^4+x^3+9x^2+6x$
- $y^2=10x^6+2x^5+6x^4+4x^3+6x^2+9x+3$
- $y^2=6x^6+8x^5+2x^4+6x^3+2x^2+8x+6$
- $y^2=3x^6+2x^5+6x^4+4x^3+6x^2+2x+3$
- $y^2=6x^6+2x^4+9x^3+8x^2+8x$
- $y^2=8x^6+8x^5+2x^4+4x^3+4x^2+7x+10$
- $y^2=2x^6+4x^5+3x^4+5x^3+9x^2+6x+10$
- $y^2=10x^6+10x^5+8x^4+3x^3+10x^2+8x+6$
- $y^2=7x^6+4x^4+2x^3+2x^2+9x+4$
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.ag $\times$ 1.11.a 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.ao $\times$ 1.121.w. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
2.11.g_w | $2$ | 2.121.i_aco |