Properties

Label 2.61.ax_jk
Base Field $\F_{61}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{61}$
Dimension:  $2$
L-polynomial:  $1 - 23 x + 244 x^{2} - 1403 x^{3} + 3721 x^{4}$
Frobenius angles:  $\pm0.109726878648$, $\pm0.321721348102$
Angle rank:  $2$ (numerical)
Number field:  4.0.512705.1
Galois group:  $D_{4}$
Jacobians:  20

This isogeny class is simple and geometrically simple.

Newton polygon

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

Point counts

This isogeny class contains the Jacobians of 20 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2540 13695680 51624778640 191751626981120 713333957925433500 2654337305764156928000 9876833019031063746159260 36751699998357337855066004480 136753057761909474758084385074960 508858111703966111284402472644152000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 39 3681 227442 13849041 844585699 51520147878 3142742990559 191707345034241 11694146513673882 713342914584741001

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
The endomorphism algebra of this simple isogeny class is 4.0.512705.1.
All geometric endomorphisms are defined over $\F_{61}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.61.x_jk$2$(not in LMFDB)