# Properties

 Label 2.169.abw_bjb 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: $1 - 48 x + 911 x^{2} - 8112 x^{3} + 28561 x^{4}$ Frobenius angles: $\pm0.0457381254353$, $\pm0.172659114562$ Angle rank: $2$ (numerical) Number field: 4.0.359568.2 Galois group: $D_{4}$ Jacobians: 6

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 6 curves, and hence is principally polarizable:

• $y^2=(3a+11)x^6+(3a+5)x^5+(4a+12)x^4+(5a+3)x^3+(3a+9)x^2+(a+3)x+5a+10$
• $y^2=(3a+2)x^6+(2a+12)x^5+(7a+3)x^4+(10a+5)x^3+(5a+9)x^2+5x+5a+2$
• $y^2=(6a+8)x^6+7ax^5+(11a+1)x^4+(10a+6)x^3+(10a+1)x^2+(8a+8)x+9a+4$
• $y^2=(7a+1)x^6+(6a+12)x^5+(3a+2)x^4+(10a+9)x^3+(10a+6)x^2+(11a+1)x+3a+1$
• $y^2=(a+10)x^6+(8a+12)x^5+6ax^4+(2a+2)x^3+(a+1)x^2+12x+3a+1$
• $y^2=(12a+1)x^6+(2a+9)x^5+9ax^4+7ax^3+(5a+11)x^2+(3a+11)x+12a+1$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21313 802072129 23280022209316 665403768037744713 19004979781804288431313 542800847477319326149144336 15502932929867310794785327216897 442779263728511907464348139507446793 12646218551674589565262864250808785020132 361188648080408976878303618099631417992316769

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 122 28080 4823066 815714980 137858607962 23298088431894 3937376418006314 665416609110550468 112455406942569159626 19004963774663884052880

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.359568.2.
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.bw_bjb $2$ (not in LMFDB)