Properties

Label 2.41.ar_fi
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:  $1 - 17 x + 138 x^{2} - 697 x^{3} + 1681 x^{4}$
Frobenius angles:  $\pm0.0660990950969$, $\pm0.386534833598$
Angle rank:  $2$ (numerical)
Number field:  4.0.4241900.1
Galois group:  $D_{4}$
Jacobians:  8

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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1106 2802604 4752389096 7977152658944 13418955038131426 22562929116813030400 37929310624670311619906 63759078234991576274991104 107178935235163539673658858216 180167781627204873407109820735084

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 25 1669 68956 2823009 115824225 4749986098 194754702265 7984931155329 327381947429596 13422659211124629

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
The endomorphism algebra of this simple isogeny class is 4.0.4241900.1.
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.r_fi$2$(not in LMFDB)