Properties

Label 2.41.as_gd
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:  $( 1 - 11 x + 41 x^{2} )( 1 - 7 x + 41 x^{2} )$
Frobenius angles:  $\pm0.171113726078$, $\pm0.315918729109$
Angle rank:  $2$ (numerical)
Jacobians:  12

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 12 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1085 2817745 4787471360 7996560250105 13424283974457125 22563522953660416000 37929231608327693590565 63759052851128198269688745 107178939921578795770045629440 180167783770694125006066333338625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 24 1676 69462 2829876 115870224 4750111118 194754296544 7984927976356 327381961744422 13422659370816476

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
The isogeny class factors as 1.41.al $\times$ 1.41.ah 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 is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.41.ae_f$2$(not in LMFDB)
2.41.e_f$2$(not in LMFDB)
2.41.s_gd$2$(not in LMFDB)