# Properties

 Label 2.41.au_gz 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: $( 1 - 11 x + 41 x^{2} )( 1 - 9 x + 41 x^{2} )$ Frobenius angles: $\pm0.171113726078$, $\pm0.251940962052$ Angle rank: $2$ (numerical) Jacobians: 9

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 9 curves, and hence is principally polarizable:

• $y^2=26x^6+3x^5+20x^4+30x^3+20x^2+3x+26$
• $y^2=31x^6+8x^5+4x^4+15x^3+4x^2+8x+31$
• $y^2=10x^6+6x^5+18x^4+15x^3+18x^2+6x+10$
• $y^2=15x^6+20x^5+19x^4+35x^3+19x^2+20x+15$
• $y^2=22x^6+14x^5+39x^4+7x^3+39x^2+14x+22$
• $y^2=30x^6+32x^5+36x^4+6x^3+36x^2+32x+30$
• $y^2=20x^6+30x^5+35x^4+5x^3+35x^2+30x+20$
• $y^2=15x^6+21x^5+6x^4+7x^3+6x^2+21x+15$
• $y^2=40x^6+23x^5+8x^4+12x^3+8x^2+23x+40$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1023 2765169 4777819200 7999636682169 13426610523770103 22564118823966720000 37929240332961803300343 63759003905183541132702249 107178921576778189769127604800 180167781230924704199688292061409

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 22 1644 69322 2830964 115890302 4750236558 194754341342 7984921846564 327381905709562 13422659181601404

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The isogeny class factors as 1.41.al $\times$ 1.41.aj 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.ac_ar $2$ (not in LMFDB) 2.41.c_ar $2$ (not in LMFDB) 2.41.u_gz $2$ (not in LMFDB)