# Properties

 Label 2.41.au_gv 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: $1 - 20 x + 177 x^{2} - 820 x^{3} + 1681 x^{4}$ Frobenius angles: $\pm0.0953421678685$, $\pm0.292668588087$ Angle rank: $2$ (numerical) Number field: 4.0.37025.1 Galois group: $D_{4}$ Jacobians: 14

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 14 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1019 2750281 4761110384 7989635062025 13422763808553979 22563165608669092096 37929143594698973338859 63759042502908302154118025 107178946498374746323041553904 180167789061443155593076886768041

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 22 1636 69082 2827428 115857102 4750035886 194753844622 7984926680388 327381981833482 13422659764981956

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The endomorphism algebra of this simple isogeny class is 4.0.37025.1.
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.u_gv $2$ (not in LMFDB)