Properties

Label 2.41.as_fw
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:  $1 - 18 x + 152 x^{2} - 738 x^{3} + 1681 x^{4}$
Frobenius angles:  $\pm0.0883036955963$, $\pm0.353631113310$
Angle rank:  $2$ (numerical)
Number field:  4.0.38720.3
Galois group:  $D_{4}$
Jacobians:  22

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 22 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1078 2792020 4761225238 7983222786000 13420342712719558 22562944141231643380 37929276269854713209158 63759096945513747017856000 107178950384549954029993308358 180167785504974562533819980360500

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 24 1662 69084 2825158 115836204 4749989262 194754525864 7984933498558 327381993703944 13422659500021902

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
The endomorphism algebra of this simple isogeny class is 4.0.38720.3.
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_fw$2$(not in LMFDB)