# Properties

 Label 2.41.au_gw 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: $( 1 - 12 x + 41 x^{2} )( 1 - 8 x + 41 x^{2} )$ Frobenius angles: $\pm0.113551764296$, $\pm0.285223287477$ Angle rank: $2$ (numerical) Jacobians: 20

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 20 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1020 2754000 4765285980 7992152064000 13423760198305500 22563439933446594000 37929192727070946323580 63759045617100820905984000 107178945262069042306587481020 180167788570021362285822821250000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 22 1638 69142 2828318 115865702 4750093638 194754096902 7984927070398 327381978057142 13422659728370598

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The isogeny class factors as 1.41.am $\times$ 1.41.ai and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ae_ao $2$ (not in LMFDB) 2.41.e_ao $2$ (not in LMFDB) 2.41.u_gw $2$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.41.ae_ao $2$ (not in LMFDB) 2.41.e_ao $2$ (not in LMFDB) 2.41.u_gw $2$ (not in LMFDB) 2.41.aw_hu $4$ (not in LMFDB) 2.41.ac_abm $4$ (not in LMFDB) 2.41.c_abm $4$ (not in LMFDB) 2.41.w_hu $4$ (not in LMFDB)