# Properties

 Label 2.41.as_fw 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: $1 - 18 x + 152 x^{2} - 738 x^{3} + 1681 x^{4}$ Frobenius angles: $\pm0.0883036955963$, $\pm0.353631113310$ Angle rank: $2$ (numerical) Number field: 4.0.38720.3 Galois group: $D_{4}$ Jacobians: 22

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 22 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1078 2792020 4761225238 7983222786000 13420342712719558 22562944141231643380 37929276269854713209158 63759096945513747017856000 107178950384549954029993308358 180167785504974562533819980360500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 24 1662 69084 2825158 115836204 4749989262 194754525864 7984933498558 327381993703944 13422659500021902

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The endomorphism algebra of this simple isogeny class is 4.0.38720.3.
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_fw $2$ (not in LMFDB)