# Properties

 Label 2.169.abs_bff 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: $1 - 44 x + 811 x^{2} - 7436 x^{3} + 28561 x^{4}$ Frobenius angles: $\pm0.0731415089542$, $\pm0.244787210186$ Angle rank: $2$ (numerical) Number field: 4.0.22196240.1 Galois group: $D_{4}$ Jacobians: 16

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 16 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21893 806822729 23295960974804 665434843684425785 19004999920929452618973 542800749495696592374649616 15502932514128958005913840551773 442779262987437940556238699180513065 12646218551952126660256965275875774357844 361188648087159859268823118553707426272315129

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 126 28248 4826370 815753076 137858754046 23298084226326 3937376312418654 665416607996851236 112455406945037134290 19004963775019100851528

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.22196240.1.
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.bs_bff $2$ (not in LMFDB)