# Properties

 Label 2.41.au_gt 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: $1 - 20 x + 175 x^{2} - 820 x^{3} + 1681 x^{4}$ Frobenius angles: $\pm0.0504478270356$, $\pm0.305285834519$ Angle rank: $2$ (numerical) Number field: 4.0.352016.1 Galois group: $D_{4}$ Jacobians: 4

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 4 curves, and hence is principally polarizable:

• $y^2=25x^6+22x^5+22x^4+3x^3+36x^2+17x+15$
• $y^2=19x^6+23x^5+x^4+35x^3+7x^2+3x+38$
• $y^2=3x^6+20x^5+7x^4+24x^3+40x^2+4x+1$
• $y^2=5x^6+32x^5+11x^4+13x^3+15x^2+29x+1$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1017 2742849 4752762372 7984567838601 13420701599035977 22562544573642827664 37928994452577705608217 63759009316294399477055625 107178937575579509291106372612 180167786067588689632148122509729

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 22 1632 68962 2825636 115839302 4749905142 194753078822 7984922524228 327381954578482 13422659541937152

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The endomorphism algebra of this simple isogeny class is 4.0.352016.1.
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.u_gt $2$ (not in LMFDB)