Properties

Label 2.41.au_gw
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:  $( 1 - 12 x + 41 x^{2} )( 1 - 8 x + 41 x^{2} )$
Frobenius angles:  $\pm0.113551764296$, $\pm0.285223287477$
Angle rank:  $2$ (numerical)
Jacobians:  20

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 20 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1020 2754000 4765285980 7992152064000 13423760198305500 22563439933446594000 37929192727070946323580 63759045617100820905984000 107178945262069042306587481020 180167788570021362285822821250000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 22 1638 69142 2828318 115865702 4750093638 194754096902 7984927070398 327381978057142 13422659728370598

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
The isogeny class factors as 1.41.am $\times$ 1.41.ai and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ae_ao$2$(not in LMFDB)
2.41.e_ao$2$(not in LMFDB)
2.41.u_gw$2$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.41.ae_ao$2$(not in LMFDB)
2.41.e_ao$2$(not in LMFDB)
2.41.u_gw$2$(not in LMFDB)
2.41.aw_hu$4$(not in LMFDB)
2.41.ac_abm$4$(not in LMFDB)
2.41.c_abm$4$(not in LMFDB)
2.41.w_hu$4$(not in LMFDB)