# Properties

 Label 2.41.as_gd 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: $( 1 - 11 x + 41 x^{2} )( 1 - 7 x + 41 x^{2} )$ Frobenius angles: $\pm0.171113726078$, $\pm0.315918729109$ Angle rank: $2$ (numerical) Jacobians: 12

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 12 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1085 2817745 4787471360 7996560250105 13424283974457125 22563522953660416000 37929231608327693590565 63759052851128198269688745 107178939921578795770045629440 180167783770694125006066333338625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 24 1676 69462 2829876 115870224 4750111118 194754296544 7984927976356 327381961744422 13422659370816476

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The isogeny class factors as 1.41.al $\times$ 1.41.ah 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 is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.41.ae_f $2$ (not in LMFDB) 2.41.e_f $2$ (not in LMFDB) 2.41.s_gd $2$ (not in LMFDB)