# Properties

 Label 2.41.at_gl Base Field $\F_{41}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

# Learn more about

## Invariants

 Base field: $\F_{41}$ Dimension: $2$ L-polynomial: $1 - 19 x + 167 x^{2} - 779 x^{3} + 1681 x^{4}$ Frobenius angles: $\pm0.127582089543$, $\pm0.309683357545$ Angle rank: $2$ (numerical) Number field: 4.0.1233477.1 Galois group: $D_{4}$ Jacobians: 6

This isogeny class is simple and geometrically 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 6 curves, and hence is principally polarizable:

• $y^2=23x^6+3x^5+38x^4+27x^3+29x^2+8x+6$
• $y^2=39x^6+14x^5+26x^4+25x^3+6x^2+22x+23$
• $y^2=7x^6+32x^5+24x^4+21x^3+6x^2+26x+13$
• $y^2=27x^6+34x^5+15x^4+35x^3+15x^2+40x+28$
• $y^2=22x^6+x^5+29x^4+6x^3+x^2+11x+14$
• $y^2=4x^6+37x^5+16x^4+32x^3+22x+40$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1051 2781997 4772407075 7992190531525 13423328697807856 22563384488596700725 37929240919197854613451 63759072754788643729282725 107178950848637129350740973675 180167787937613187857793841837312

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 23 1655 69245 2828331 115861978 4750081967 194754344353 7984930469011 327381995121515 13422659681255630

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The endomorphism algebra of this simple isogeny class is 4.0.1233477.1.
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.t_gl $2$ (not in LMFDB)