# Properties

 Label 2.41.as_fx Base Field $\F_{41}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{41}$ Dimension: $2$ L-polynomial: $1 - 18 x + 153 x^{2} - 738 x^{3} + 1681 x^{4}$ Frobenius angles: $\pm0.101373436493$, $\pm0.349335512379$ Angle rank: $2$ (numerical) Number field: 4.0.3342400.2 Galois group: $D_{4}$ Jacobians: 6

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## Point counts

This isogeny class contains the Jacobians of 6 curves, and hence is principally polarizable:

• $y^2=35x^6+x^5+4x^4+16x^3+9x^2+25x+14$
• $y^2=13x^6+29x^5+6x^4+31x^3+39x^2+37x+30$
• $y^2=38x^6+13x^5+40x^3+9x^2+9x+11$
• $y^2=14x^6+39x^5+39x^4+19x^3+26x^2+23x+12$
• $y^2=22x^6+39x^4+x^3+28x^2+21x+1$
• $y^2=24x^6+7x^5+26x^4+16x^3+5x^2+6x+28$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1079 2795689 4764971900 7985161545049 13420968266033159 22563083485887552400 37929303017347036212359 63759104377295843693028969 107178953023456159915347203900 180167786128645038389263878156409

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 24 1664 69138 2825844 115841604 4750018598 194754663204 7984934429284 327382001764578 13422659546485904

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The endomorphism algebra of this simple isogeny class is 4.0.3342400.2.
All geometric endomorphisms are defined over $\F_{41}$.

## 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 2.41.s_fx $2$ (not in LMFDB)