Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 7 x + 29 x^{2} - 77 x^{3} + 121 x^{4}$ |
Frobenius angles: | $\pm0.162126013132$, $\pm0.441671623734$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.196245.1 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
Isomorphism classes: | 2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $67$ | $15745$ | $1820725$ | $213329005$ | $25941832912$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $131$ | $1367$ | $14571$ | $161080$ | $1775423$ | $19503685$ | $214373251$ | $2357863307$ | $25937221526$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=8x^6+2x^5+5x^4+5x^3+5x^2+6x+6$
- $y^2=2x^6+9x^5+10x^4+8x^3+3x^2+10x+1$
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 4.0.196245.1. |
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.h_bd | $2$ | 2.121.j_f |