Properties

Label 2.61.aba_kz
Base field $\F_{61}$
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:  $\F_{61}$
Dimension:  $2$
L-polynomial:  $1 - 26 x + 285 x^{2} - 1586 x^{3} + 3721 x^{4}$
Frobenius angles:  $\pm0.0471438380502$, $\pm0.263959254932$
Angle rank:  $2$ (numerical)
Number field:  4.0.406080.2
Galois group:  $D_{4}$
Jacobians:  $4$
Isomorphism classes:  4

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $2395$ $13457505$ $51496571500$ $191723273550345$ $713333008278344875$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $36$ $3616$ $226878$ $13846996$ $844584576$ $51519970078$ $3142737883776$ $191707276568356$ $11694145959510198$ $713342912228782576$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61}$.

Endomorphism algebra over $\F_{61}$
The endomorphism algebra of this simple isogeny class is 4.0.406080.2.

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.ba_kz$2$(not in LMFDB)