Properties

Label 2.41.as_gf
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:  $1 - 18 x + 161 x^{2} - 738 x^{3} + 1681 x^{4}$
Frobenius angles:  $\pm0.197716147647$, $\pm0.298200122822$
Angle rank:  $2$ (numerical)
Number field:  4.0.4672.2
Galois group:  $D_{4}$
Jacobians:  9

This isogeny class is simple and geometrically 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
$A(\F_{q^r})$ 1087 2825113 4794978748 8000270823033 13425222525857407 22563519729022173328 37929122998774759264543 63759003583246978501382697 107178928319344659924873508348 180167782959452249414726279823433

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 24 1680 69570 2831188 115878324 4750110438 194753738868 7984921806244 327381926304978 13422659310378240

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
The endomorphism algebra of this simple isogeny class is 4.0.4672.2.
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.s_gf$2$(not in LMFDB)