Properties

Label 2.41.au_gz
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:  $( 1 - 11 x + 41 x^{2} )( 1 - 9 x + 41 x^{2} )$
Frobenius angles:  $\pm0.171113726078$, $\pm0.251940962052$
Angle rank:  $2$ (numerical)
Jacobians:  9

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 9 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1023 2765169 4777819200 7999636682169 13426610523770103 22564118823966720000 37929240332961803300343 63759003905183541132702249 107178921576778189769127604800 180167781230924704199688292061409

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 22 1644 69322 2830964 115890302 4750236558 194754341342 7984921846564 327381905709562 13422659181601404

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
The isogeny class factors as 1.41.al $\times$ 1.41.aj 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.ac_ar$2$(not in LMFDB)
2.41.c_ar$2$(not in LMFDB)
2.41.u_gz$2$(not in LMFDB)