# Properties

 Label 2.41.ay_is 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: $( 1 - 12 x + 41 x^{2} )^{2}$ Frobenius angles: $\pm0.113551764296$, $\pm0.113551764296$ Angle rank: $1$ (numerical) Jacobians: 3

This isogeny class is not 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:

• $y^2=8x^6+17x^4+30x^3+35x^2+36$
• $y^2=29x^6+26x^4+26x^2+29$
• $y^2=27x^6+10x^5+9x^4+19x^3+x^2+31x+38$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 900 2624400 4715568900 7982207078400 13423713390562500 22564196541890816400 37929502028360121480900 63759117458480055518822400 107178954605339646251452752900 180167788621537529528113580250000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 18 1558 68418 2824798 115865298 4750252918 194755685058 7984936067518 327382006596498 13422659732208598

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The isogeny class factors as 1.41.am 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-5})$$$)$
All geometric endomorphisms are defined over $\F_{41}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.41.a_ack $2$ (not in LMFDB) 2.41.y_is $2$ (not in LMFDB) 2.41.m_dz $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.41.a_ack $2$ (not in LMFDB) 2.41.y_is $2$ (not in LMFDB) 2.41.m_dz $3$ (not in LMFDB) 2.41.a_ck $4$ (not in LMFDB) 2.41.am_dz $6$ (not in LMFDB)