# Properties

 Label 2.41.as_gh 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: $( 1 - 9 x + 41 x^{2} )^{2}$ Frobenius angles: $\pm0.251940962052$, $\pm0.251940962052$ Angle rank: $1$ (numerical) Jacobians: 15

This isogeny class is not 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 15 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1089 2832489 4802490000 8003936949129 13426077749128209 22563442409840640000 37928973949344795906369 63758940768104991107368329 107178915157433098275355290000 180167783335487620439204765476809

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 24 1684 69678 2832484 115885704 4750094158 194752973544 7984913939524 327381886101438 13422659338393204

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The isogeny class factors as 1.41.aj 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-83})$$$)$
All geometric endomorphisms are defined over $\F_{41}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.41.a_b $2$ (not in LMFDB) 2.41.s_gh $2$ (not in LMFDB) 2.41.j_bo $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.41.a_b $2$ (not in LMFDB) 2.41.s_gh $2$ (not in LMFDB) 2.41.j_bo $3$ (not in LMFDB) 2.41.a_ab $4$ (not in LMFDB) 2.41.aj_bo $6$ (not in LMFDB)