Properties

Label 2.41.as_gc
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:  $1 - 18 x + 158 x^{2} - 738 x^{3} + 1681 x^{4}$
Frobenius angles:  $\pm0.159283075817$, $\pm0.322876743820$
Angle rank:  $2$ (numerical)
Number field:  4.0.111600.3
Galois group:  $D_{4}$
Jacobians:  24

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 24 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1084 2814064 4783719100 7994688286464 13423783447978684 22563496637553295600 37929270380008636703164 63759071937483269494861824 107178944755958165574442479100 180167784403117205421784853119984

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 24 1674 69408 2829214 115865904 4750105578 194754495624 7984930366654 327381976511208 13422659417932554

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
The endomorphism algebra of this simple isogeny class is 4.0.111600.3.
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_gc$2$(not in LMFDB)