# Properties

 Label 2.41.ar_fi 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: $1 - 17 x + 138 x^{2} - 697 x^{3} + 1681 x^{4}$ Frobenius angles: $\pm0.0660990950969$, $\pm0.386534833598$ Angle rank: $2$ (numerical) Number field: 4.0.4241900.1 Galois group: $D_{4}$ Jacobians: 8

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 8 curves, and hence is principally polarizable:

• $y^2=35x^6+34x^5+9x^4+18x^3+32x^2+28x+18$
• $y^2=24x^6+x^5+6x^4+4x^3+14x^2+36x+35$
• $y^2=9x^6+38x^5+15x^4+33x^2+24x+17$
• $y^2=24x^6+37x^5+3x^4+24x^3+13x^2+34x+10$
• $y^2=17x^6+31x^5+38x^4+23x^3+29x^2+11x+3$
• $y^2=28x^6+36x^5+34x^4+10x^3+29x+26$
• $y^2=7x^6+31x^5+11x^4+23x^3+13x^2+30x+1$
• $y^2=24x^6+34x^5+21x^3+5x^2+9x+29$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1106 2802604 4752389096 7977152658944 13418955038131426 22562929116813030400 37929310624670311619906 63759078234991576274991104 107178935235163539673658858216 180167781627204873407109820735084

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 25 1669 68956 2823009 115824225 4749986098 194754702265 7984931155329 327381947429596 13422659211124629

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The endomorphism algebra of this simple isogeny class is 4.0.4241900.1.
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.r_fi $2$ (not in LMFDB)