# Properties

 Label 2.41.as_gc 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: $1 - 18 x + 158 x^{2} - 738 x^{3} + 1681 x^{4}$ Frobenius angles: $\pm0.159283075817$, $\pm0.322876743820$ Angle rank: $2$ (numerical) Number field: 4.0.111600.3 Galois group: $D_{4}$ Jacobians: 24

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 24 curves, and hence is principally polarizable:

• $y^2=29x^6+6x^5+15x^4+15x^2+17x+34$
• $y^2=17x^6+39x^5+10x^3+x^2+37x+10$
• $y^2=6x^6+22x^5+15x^4+14x^3+35x^2+40x+7$
• $y^2=37x^6+36x^5+40x^4+28x^3+26x^2+18x+24$
• $y^2=38x^6+37x^5+38x^4+9x^3+18x^2+32x+31$
• $y^2=38x^6+25x^5+34x^4+18x^3+18x^2+36x+17$
• $y^2=28x^6+34x^5+27x^4+34x^2+24x+11$
• $y^2=6x^6+39x^5+26x^4+11x^3+37x^2+15x+11$
• $y^2=8x^6+16x^5+6x^4+26x^3+9x^2+12x+7$
• $y^2=14x^6+25x^5+35x^4+24x^3+39x^2+3x+31$
• $y^2=32x^6+13x^5+20x^4+6x^3+11x^2+7x+15$
• $y^2=30x^6+39x^5+20x^4+35x^3+14x^2+18x+35$
• $y^2=15x^6+9x^5+21x^4+21x^3+5x^2+27x+38$
• $y^2=37x^6+7x^5+31x^4+17x^3+15x^2+27x+27$
• $y^2=12x^6+25x^5+21x^4+9x^3+26x^2+23x+13$
• $y^2=13x^6+9x^5+4x^4+21x^3+9x^2+26x+24$
• $y^2=8x^6+3x^5+6x^4+17x^3+21x^2+33x+30$
• $y^2=35x^6+21x^5+24x^4+17x^3+2x^2+4x+30$
• $y^2=3x^6+4x^5+25x^4+29x^3+39x^2+35x+14$
• $y^2=14x^6+25x^5+24x^4+7x^3+11x^2+8x+19$
• $y^2=11x^6+34x^5+32x^4+23x^3+22x^2+14x+17$
• $y^2=x^6+25x^5+38x^4+11x^3+33x^2+30x+32$
• $y^2=27x^6+14x^5+28x^4+40x^3+30x^2+13x+1$
• $y^2=39x^6+21x^5+14x^4+28x^3+32x^2+11x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1084 2814064 4783719100 7994688286464 13423783447978684 22563496637553295600 37929270380008636703164 63759071937483269494861824 107178944755958165574442479100 180167784403117205421784853119984

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 24 1674 69408 2829214 115865904 4750105578 194754495624 7984930366654 327381976511208 13422659417932554

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The endomorphism algebra of this simple isogeny class is 4.0.111600.3.
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_gc $2$ (not in LMFDB)