Properties

Label 2.13.ah_bf
Base field $\F_{13}$
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:  $\F_{13}$
Dimension:  $2$
L-polynomial:  $1 - 7 x + 31 x^{2} - 91 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.171237405357$, $\pm0.464284369344$
Angle rank:  $2$ (numerical)
Number field:  4.0.589541.1
Galois group:  $D_{4}$
Jacobians:  $3$
Isomorphism classes:  3

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $103$ $30797$ $4908259$ $812332469$ $138024863248$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $7$ $183$ $2233$ $28443$ $371742$ $4834623$ $62772577$ $815717811$ $10604298439$ $137858354518$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13}$.

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 4.0.589541.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.13.h_bf$2$2.169.n_z