# Properties

 Label 2.169.abt_bgf 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: $1 - 45 x + 837 x^{2} - 7605 x^{3} + 28561 x^{4}$ Frobenius angles: $\pm0.0795349314068$, $\pm0.224304217515$ Angle rank: $2$ (numerical) Number field: 4.0.9860725.1 Galois group: $D_{4}$ Jacobians: 40

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 40 curves, and hence is principally polarizable:

• $y^2=(2a+7)x^6+(6a+10)x^5+(5a+7)x^4+(4a+1)x^3+(a+10)x^2+(3a+10)x+7a+6$
• $y^2=(6a+4)x^6+(8a+1)x^5+(a+6)x^4+(6a+6)x^3+(6a+7)x^2+(10a+8)x+5a+2$
• $y^2=(6a+1)x^6+(9a+1)x^5+(5a+11)x^4+(12a+3)x^3+(3a+5)x^2+(7a+2)x+a$
• $y^2=(11a+2)x^6+(2a+4)x^5+(9a+12)x^4+4ax^3+(8a+3)x^2+(4a+9)x+7a+5$
• $y^2=(3a+6)x^6+(12a+1)x^5+(9a+5)x^4+(11a+10)x^3+9ax^2+(12a+9)x+10a+6$
• $y^2=(10a+1)x^6+(4a+4)x^5+(6a+8)x^4+(2a+5)x^3+(2a+1)x^2+(9a+6)x+2$
• $y^2=(6a+12)x^6+5ax^5+9ax^4+(3a+2)x^3+(8a+3)x^2+(a+8)x+4a+8$
• $y^2=(12a+4)x^6+(3a+1)x^5+(12a+10)x^4+(9a+6)x^3+ax^2+(6a+10)x+11a+6$
• $y^2=(7a+1)x^6+(9a+11)x^5+(a+8)x^4+12x^3+(3a+4)x^2+(12a+7)x+10a$
• $y^2=(11a+1)x^6+6ax^5+(3a+8)x^4+(10a+10)x^3+(a+11)x^2+(7a+8)x+4a+6$
• $y^2=(6a+10)x^6+(8a+7)x^5+(11a+10)x^4+(12a+6)x^3+(3a+10)x^2+(5a+9)x+12a+1$
• $y^2=(3a+5)x^6+(2a+1)x^5+3ax^4+(11a+3)x^3+(9a+1)x^2+(5a+6)x+3a+6$
• $y^2=(5a+10)x^6+(5a+1)x^5+(7a+6)x^4+(5a+4)x^3+(12a+4)x^2+(8a+2)x+7a+5$
• $y^2=(2a+5)x^6+(5a+5)x^5+(a+1)x^4+(12a+2)x^2+(a+8)x+7a+5$
• $y^2=6ax^6+6ax^5+(12a+12)x^4+(12a+7)x^3+(10a+7)x^2+10x+a+3$
• $y^2=(12a+6)x^6+(7a+11)x^5+(9a+9)x^4+(a+12)x^3+(3a+1)x^2+(11a+3)x+7a+10$
• $y^2=(6a+12)x^6+(12a+8)x^5+(11a+3)x^4+(9a+1)x^3+(7a+2)x^2+(10a+5)x+8a+7$
• $y^2=(11a+8)x^6+(11a+11)x^5+(12a+9)x^4+(a+10)x^3+(6a+10)x^2+(a+4)x+7a+9$
• $y^2=(12a+6)x^6+(11a+6)x^5+9x^4+(9a+2)x^3+(a+12)x^2+(10a+2)x+2a+10$
• $y^2=(5a+8)x^6+(10a+7)x^5+4ax^4+(7a+11)x^3+(a+3)x^2+10ax+5a+9$
• and 20 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21749 805778701 23293526135789 665435600168336725 19005026431679376465104 542800859011722574225492621 15502932782296934417995655463269 442779263360576723244600296641642725 12646218551845414665141604606144941337709 361188648084939903511598165800956803748130816

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 125 28211 4825865 815754003 137858946350 23298088926971 3937376380526945 665416608557610883 112455406944088207085 19004963774902291591406

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.9860725.1.
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.bt_bgf $2$ (not in LMFDB)