# Properties

 Label 2.41.au_ha 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: $( 1 - 10 x + 41 x^{2} )^{2}$ Frobenius angles: $\pm0.214776712523$, $\pm0.214776712523$ Angle rank: $1$ (numerical) Jacobians: 10

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 10 curves, and hence is principally polarizable:

• $y^2=5x^6+24x^4+24x^2+5$
• $y^2=27x^6+40x^5+36x^4+14x^3+36x^2+40x+27$
• $y^2=38x^5+22x^4+13x^3+12x^2+13x$
• $y^2=12x^6+30x^4+30x^2+12$
• $y^2=35x^6+26x^5+26x^4+27x^3+14x^2+23x+7$
• $y^2=29x^6+27x^4+27x^2+29$
• $y^2=3x^6+23x^5+32x^4+11x^3+32x^2+23x+3$
• $y^2=34x^6+8x^5+37x^4+11x^3+31x^2+9x+29$
• $y^2=31x^6+31x^5+2x^4+6x^3+25x^2+16x+36$
• $y^2=13x^6+13x^5+30x^4+2x^3+30x^2+13x+13$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1024 2768896 4781999104 8002109440000 13427514355631104 22564297283790585856 37929223483288697979904 63758973775802534461440000 107178907796429068502685942784 180167777394336803493908044988416

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 22 1646 69382 2831838 115898102 4750274126 194754254822 7984918073278 327381863616982 13422658895772206

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The isogeny class factors as 1.41.ak 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$
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_as $2$ (not in LMFDB) 2.41.u_ha $2$ (not in LMFDB) 2.41.k_ch $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.41.a_as $2$ (not in LMFDB) 2.41.u_ha $2$ (not in LMFDB) 2.41.k_ch $3$ (not in LMFDB) 2.41.as_gg $4$ (not in LMFDB) 2.41.aq_fq $4$ (not in LMFDB) 2.41.ac_c $4$ (not in LMFDB) 2.41.a_s $4$ (not in LMFDB) 2.41.c_c $4$ (not in LMFDB) 2.41.q_fq $4$ (not in LMFDB) 2.41.s_gg $4$ (not in LMFDB) 2.41.ak_ch $6$ (not in LMFDB) 2.41.a_adc $8$ (not in LMFDB) 2.41.a_dc $8$ (not in LMFDB) 2.41.ai_x $12$ (not in LMFDB) 2.41.i_x $12$ (not in LMFDB)