# Properties

 Label 2.41.at_gk 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: $( 1 - 12 x + 41 x^{2} )( 1 - 7 x + 41 x^{2} )$ Frobenius angles: $\pm0.113551764296$, $\pm0.315918729109$ Angle rank: $2$ (numerical) Jacobians: 10

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 10 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1050 2778300 4768444800 7989990724800 13422569265656250 22563223610314060800 37929229263299849875050 63759078059195325925036800 107178953226187440605312188800 180167788518282043328992436437500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 23 1653 69188 2827553 115855423 4750048098 194754284503 7984931133313 327382002383828 13422659724515973

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The isogeny class factors as 1.41.am $\times$ 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.af_ac $2$ (not in LMFDB) 2.41.f_ac $2$ (not in LMFDB) 2.41.t_gk $2$ (not in LMFDB)