Properties

Label 2.13.aj_bq
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:  $1 - 9 x + 42 x^{2} - 117 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.136139978944$, $\pm0.390198274089$
Angle rank:  $2$ (numerical)
Number field:  4.0.119068.1
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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $86$ $29068$ $4947752$ $815299264$ $137654604926$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $5$ $173$ $2252$ $28545$ $370745$ $4828394$ $62774717$ $815838625$ $10604648252$ $137858097413$

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 $\F_{13}$.

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 4.0.119068.1.

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.j_bq$2$2.169.d_ae