Properties

Label 2.13.aj_bs
Base Field $\F_{13}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{13}$
Dimension:  $2$
L-polynomial:  $( 1 - 6 x + 13 x^{2} )( 1 - 3 x + 13 x^{2} )$
Frobenius angles:  $\pm0.187167041811$, $\pm0.363422825076$
Angle rank:  $2$ (numerical)
Jacobians:  3

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class contains the Jacobians of 3 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 88 29920 5070208 823996800 137921504248 23299836651520 3938071721433784 665461638360921600 112455650903749808512 19004827709997678181600

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 177 2306 28849 371465 4827174 62759597 815785921 10604522378 137857504857

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ag $\times$ 1.13.ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{13}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.13.ad_i$2$2.169.h_gm
2.13.d_i$2$2.169.h_gm
2.13.j_bs$2$2.169.h_gm
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.13.ad_i$2$2.169.h_gm
2.13.d_i$2$2.169.h_gm
2.13.j_bs$2$2.169.h_gm
2.13.ah_bm$4$(not in LMFDB)
2.13.ab_o$4$(not in LMFDB)
2.13.b_o$4$(not in LMFDB)
2.13.h_bm$4$(not in LMFDB)