# Properties

 Label 2.41.as_fv 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: $1 - 18 x + 151 x^{2} - 738 x^{3} + 1681 x^{4}$ Frobenius angles: $\pm0.0737449254048$, $\pm0.357708845091$ Angle rank: $2$ (numerical) Number field: 4.0.166032.2 Galois group: $D_{4}$ Jacobians: 12

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 12 curves, and hence is principally polarizable:

• $y^2=16x^6+37x^5+25x^4+24x^3+30x^2+2x+27$
• $y^2=14x^6+7x^5+x^4+27x^3+35x^2+18x+24$
• $y^2=9x^6+19x^5+40x^4+16x^3+12x^2+23x+21$
• $y^2=11x^6+27x^5+23x^4+28x^3+14x^2+35x+3$
• $y^2=31x^6+26x^5+40x^4+10x^3+31x^2+9x+15$
• $y^2=7x^6+7x^5+16x^4+39x^3+13x^2+7$
• $y^2=35x^6+9x^5+4x^4+3x^3+24x^2+9x+34$
• $y^2=38x^6+14x^5+23x^4+23x^3+16x^2+5x+14$
• $y^2=22x^6+27x^5+19x^4+8x^3+9x^2+28x+12$
• $y^2=40x^6+x^5+15x^4+x^3+36x^2+37x+28$
• $y^2=22x^6+34x^5+19x^4+27x^3+15x^2+37x+29$
• $y^2=36x^6+18x^5+5x^4+22x^3+36x^2+5x+20$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1077 2788353 4757479488 7981272881433 13419696318179757 22562785758466003968 37929238087133082166413 63759084431578340749669737 107178945938393199920974352448 180167784321373025185937304447633

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 24 1660 69030 2824468 115830624 4749955918 194754329808 7984931931364 327381980122998 13422659411842540

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The endomorphism algebra of this simple isogeny class is 4.0.166032.2.
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_fv $2$ (not in LMFDB)