Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 6 x + 28 x^{2} - 66 x^{3} + 121 x^{4}$ |
Frobenius angles: | $\pm0.247161792509$, $\pm0.438778135579$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.131904.1 |
Galois group: | $D_{4}$ |
Jacobians: | $10$ |
Isomorphism classes: | 10 |
This isogeny class is simple and 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $78$ | $17316$ | $1896102$ | $215341776$ | $25926727398$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $142$ | $1422$ | $14710$ | $160986$ | $1772782$ | $19489938$ | $214328734$ | $2357777862$ | $25937284702$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=8x^6+5x^5+x^4+x^3+10x^2+2x+8$
- $y^2=8x^6+x^5+6x^4+3x^3+7x^2+4x$
- $y^2=4x^6+x^4+6x^3+10x^2+4x+3$
- $y^2=4x^6+4x^5+9x^4+5x^3+10x^2+3x+8$
- $y^2=3x^6+2x^5+5x^4+7x^3+5x^2+9x+7$
- $y^2=2x^6+5x^5+3x^4+8x^3+3x^2+9x+10$
- $y^2=7x^6+7x^5+7x^4+x^3+5x^2+10$
- $y^2=5x^6+8x^5+8x^4+8x+6$
- $y^2=7x^6+3x^4+9x^3+9x^2+3x+8$
- $y^2=9x^6+3x^5+8x^4+5x^3+10x^2+7x+7$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11}$.
Endomorphism algebra over $\F_{11}$The endomorphism algebra of this simple isogeny class is 4.0.131904.1. |
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_bc | $2$ | 2.121.u_ja |