Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 2 x + 6 x^{2} - 22 x^{3} + 121 x^{4}$ |
Frobenius angles: | $\pm0.219087573611$, $\pm0.656044000493$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.43928.1 |
Galois group: | $D_{4}$ |
Jacobians: | $20$ |
Isomorphism classes: | 32 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $104$ | $15808$ | $1721096$ | $219035648$ | $26143963144$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $130$ | $1294$ | $14958$ | $162330$ | $1770418$ | $19488766$ | $214358814$ | $2357758186$ | $25937219490$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 20 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+5x^5+9x^4+8x^3+5x^2+3x$
- $y^2=10x^6+5x^5+10x^4+3x^3+2x^2+2x+1$
- $y^2=x^6+8x^5+x^4+2x^3+3x^2+9x+2$
- $y^2=6x^6+10x^4+9x^3+6x^2+10x+7$
- $y^2=6x^6+5x^5+7x^4+x^3+4x^2+3x+3$
- $y^2=5x^6+10x^5+6x^4+x^3+2x^2+4x$
- $y^2=4x^6+x^5+10x^3+6x^2+6x$
- $y^2=6x^6+7x^5+x^4+5x^2+4x+7$
- $y^2=2x^5+4x^4+10x^3+3x^2+5x$
- $y^2=9x^6+x^5+5x^4+6x^3+7x^2+9x+2$
- $y^2=6x^6+2x^5+2x^4+7x^3+9x^2+x+1$
- $y^2=5x^6+10x^5+10x^4+8x^3+x^2+1$
- $y^2=4x^6+x^5+9x^4+6x^3+8x^2+x$
- $y^2=3x^5+2x^4+8x^3+2x^2+10x+3$
- $y^2=5x^6+10x^5+3x^4+5x^2+3x+9$
- $y^2=x^6+7x^5+6x^4+x^3+2x^2+8x+2$
- $y^2=10x^6+2x^5+7x^4+2x^3+6x^2+6x$
- $y^2=6x^5+5x^4+3x+6$
- $y^2=5x^6+6x^5+10x^4+3x^3+10x^2+x+2$
- $y^2=3x^6+6x^5+8x^4+5x^3+5x^2+9x+1$
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 4.0.43928.1. |
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.c_g | $2$ | 2.121.i_hi |