Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 5 x + 18 x^{2} - 55 x^{3} + 121 x^{4}$ |
Frobenius angles: | $\pm0.170748385587$, $\pm0.533728959213$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.8405.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $80$ | $16000$ | $1747520$ | $213056000$ | $26119222000$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $7$ | $133$ | $1312$ | $14553$ | $162177$ | $1776358$ | $19488427$ | $214351473$ | $2358015712$ | $25937386773$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=7x^6+2x^5+9x^4+x^3+x^2+x+1$
- $y^2=2x^6+4x^5+10x^4+6x^3+7x^2+8x+6$
- $y^2=6x^6+8x^5+9x^4+x^3+3x^2+10x+8$
- $y^2=7x^6+10x^5+7x^4+4x^3+9x^2+10x+7$
- $y^2=6x^6+8x^5+7x^4+x^3+4x^2+x+6$
- $y^2=3x^6+10x^5+7x^4+4x^3+3x^2+4x+10$
- $y^2=7x^6+x^5+2x^4+8x^3+3x^2+4x+8$
- $y^2=10x^6+8x^5+x^3+7x^2+2x+5$
- $y^2=10x^6+8x^5+2x^4+4x^2+10x+6$
- $y^2=10x^6+5x^5+4x^4+8x^3+x^2+5$
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 4.0.8405.1. |
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.f_s | $2$ | 2.121.l_q |