Properties

Label 2.41.au_ha
Base Field $\F_{41}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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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:

Point counts of the abelian variety

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

Point counts of the curve

$r$ 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.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)