Properties

Label 2.41.ar_fi
Base field $\F_{41}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
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:  $8$
Isomorphism classes:  16

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1106$ $2802604$ $4752389096$ $7977152658944$ $13418955038131426$

Point counts of the curve

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

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{41}$.

Endomorphism algebra over $\F_{41}$
The endomorphism algebra of this simple isogeny class is 4.0.4241900.1.

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)