Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 6 x + 24 x^{2} - 66 x^{3} + 121 x^{4}$ |
Frobenius angles: | $\pm0.175918430288$, $\pm0.482992569757$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.417088.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $74$ | $16132$ | $1798274$ | $213006928$ | $26023590434$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $134$ | $1350$ | $14550$ | $161586$ | $1776710$ | $19496994$ | $214340638$ | $2357877654$ | $25937467574$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^5+5x^3+10x^2+8x$
- $y^2=8x^6+10x^5+x^4+3x^3+7x^2+2x+8$
- $y^2=5x^6+4x^5+4x^4+10x^3+10x^2+9x+6$
- $y^2=6x^6+9x^5+3x^4+5x^3+3x+6$
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 4.0.417088.1. |
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_y | $2$ | 2.121.m_ba |