# Properties

 Label 2.41.aw_ht 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: $1 - 22 x + 201 x^{2} - 902 x^{3} + 1681 x^{4}$ Frobenius angles: $\pm0.0789642873101$, $\pm0.230762790725$ Angle rank: $2$ (numerical) Number field: 4.0.45632.1 Galois group: $D_{4}$ Jacobians: 2

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 2 curves, and hence is principally polarizable:

• $y^2=19x^6+22x^5+12x^4+31x^3+4x^2+38x+11$
• $y^2=11x^6+21x^5+4x^4+21x^3+18x^2+31x+6$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 959 2691913 4744061756 7988954416793 13424058719930719 22563668704226881168 37929194619491139697439 63759009104968249758154217 107178926745281611659133934204 180167783644507226540139088943833

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 20 1600 68834 2827188 115868280 4750141798 194754106616 7984922497764 327381921496946 13422659361415440

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The endomorphism algebra of this simple isogeny class is 4.0.45632.1.
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.w_ht $2$ (not in LMFDB)