# Properties

 Label 2.41.at_gi Base Field $\F_{41}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{41}$ Dimension: $2$ L-polynomial: $1 - 19 x + 164 x^{2} - 779 x^{3} + 1681 x^{4}$ Frobenius angles: $\pm0.0831029995597$, $\pm0.326848656985$ Angle rank: $2$ (numerical) Number field: 4.0.357192.2 Galois group: $D_{4}$ Jacobians: 6

This isogeny class is simple and geometrically simple.

## Newton polygon

 $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1]$

## Point counts

This isogeny class contains the Jacobians of 6 curves, and hence is principally polarizable:

• $y^2=22x^6+34x^5+15x^4+6x^3+37x^2+8x+18$
• $y^2=28x^6+13x^5+16x^4+5x^3+2x^2+15x+15$
• $y^2=9x^6+35x^5+20x^4+3x^3+7x^2+35x+22$
• $y^2=12x^6+31x^5+30x^4+27x^3+4x^2+6x+7$
• $y^2=35x^6+31x^5+12x^4+40x^3+30x^2+5x+24$
• $y^2=31x^6+39x^5+9x^4+24x^3+22x^2+35x+32$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1048 2770912 4760523232 7985557794688 13420984426801048 22562837588624171008 37929164611481170118488 63759069174851069175111168 107178950938488724582675057888 180167787706485985312147331307232

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 23 1649 69074 2825985 115841743 4749966830 194753952535 7984930020673 327381995395970 13422659664036449

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The endomorphism algebra of this simple isogeny class is 4.0.357192.2.
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_gi $2$ (not in LMFDB)