# Properties

 Label 2.169.abx_bkb Base Field $\F_{13^{2}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{13^{2}}$ Dimension: $2$ L-polynomial: $1 - 49 x + 937 x^{2} - 8281 x^{3} + 28561 x^{4}$ Frobenius angles: $\pm0.0546290859232$, $\pm0.144071978349$ Angle rank: $2$ (numerical) Number field: 4.0.63725.1 Galois group: $D_{4}$ Jacobians: 3

This isogeny class is simple and geometrically 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:

• $y^2=(4a+3)x^6+(5a+7)x^5+(10a+12)x^4+(10a+9)x^3+(7a+10)x^2+(6a+5)x+9a$
• $y^2=(10a+10)x^6+(a+1)x^5+(7a+11)x^4+(11a+2)x^3+(3a+5)x^2+(9a+1)x+9a$
• $y^2=(3a+9)x^6+(2a+5)x^5+(3a+11)x^4+(12a+10)x^3+(10a+1)x^2+(11a+10)x+4a+11$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21169 800802101 23275156499641 665391630093649349 19004962180730770132624 542800859353697857490675141 15502933118188460997199657976089 442779264526745585934223215518376389 12646218554090026251167863159276894578721 361188648086261033867669145852153977938968576

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 121 28035 4822057 815700099 137858480286 23298088941651 3937376465835409 665416610310150339 112455406964048221873 19004963774971806607150

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.63725.1.
All geometric endomorphisms are defined over $\F_{13^{2}}$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.169.bx_bkb $2$ (not in LMFDB)