# Properties

 Label 2.41.av_hh 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: $1 - 21 x + 189 x^{2} - 861 x^{3} + 1681 x^{4}$ Frobenius angles: $\pm0.0895520221991$, $\pm0.262353290689$ Angle rank: $2$ (numerical) Number field: 4.0.188773.1 Galois group: $D_{4}$ Jacobians: 3

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 3 curves, and hence is principally polarizable:

• $y^2=40x^6+33x^5+31x^4+16x^3+28x^2+39x+11$
• $y^2=13x^6+13x^5+15x^4+8x^3+30x^2+7x+14$
• $y^2=12x^6+36x^5+25x^4+8x^3+34x^2+10x+9$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 989 2722717 4754440469 7990218721333 13423642423672784 22563415794468383293 37929144199923688788701 63759016316936981557022373 107178935676419235326651791301 180167787077539419765035434258432

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 21 1619 68985 2827635 115864686 4750088555 194753847729 7984923400963 327381948777429 13422659617179374

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The endomorphism algebra of this simple isogeny class is 4.0.188773.1.
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.v_hh $2$ (not in LMFDB)