# Properties

 Label 2.169.abs_bfc 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: $1 - 44 x + 808 x^{2} - 7436 x^{3} + 28561 x^{4}$ Frobenius angles: $\pm0.0449093466818$, $\pm0.252181585824$ Angle rank: $2$ (numerical) Number field: 4.0.14362880.1 Galois group: $D_{4}$ Jacobians: 48

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 48 curves, and hence is principally polarizable:

• $y^2=(7a+3)x^6+(11a+10)x^5+(10a+6)x^4+(7a+2)x^3+(4a+12)x^2+7x+9a+1$
• $y^2=(10a+9)x^6+(a+10)x^5+(4a+3)x^4+(9a+5)x^3+5x^2+(9a+3)x+8a+5$
• $y^2=(10a+7)x^6+2ax^5+(4a+9)x^4+(2a+2)x^3+(a+3)x^2+(10a+9)x+a+9$
• $y^2=5ax^6+(4a+4)x^5+(10a+9)x^4+(12a+5)x^3+(12a+7)x^2+(a+11)x+4a+10$
• $y^2=10ax^6+(8a+11)x^5+2x^4+(2a+1)x^3+(12a+12)x^2+(6a+8)x+a+3$
• $y^2=(2a+3)x^6+(a+2)x^5+11ax^4+(9a+4)x^3+(12a+8)x^2+3x+3a+6$
• $y^2=(9a+4)x^6+(7a+11)x^5+(7a+8)x^4+(3a+7)x^3+(10a+1)x^2+(5a+2)x+3a+9$
• $y^2=(7a+12)x^6+(8a+12)x^5+(5a+6)x^4+(7a+11)x^3+(9a+8)x^2+(10a+9)x+9a+2$
• $y^2=(2a+11)x^6+(2a+2)x^5+(7a+8)x^4+(a+3)x^3+(10a+4)x^2+6x+10a+2$
• $y^2=(3a+10)x^6+(2a+1)x^5+(2a+8)x^4+(2a+12)x^3+(11a+4)x^2+(9a+6)x+8a+11$
• $y^2=(6a+2)x^6+9x^5+(11a+11)x^4+(5a+4)x^3+(5a+12)x^2+(8a+9)x+8a+12$
• $y^2=(5a+7)x^6+(5a+11)x^5+(7a+8)x^4+(10a+12)x^3+6ax^2+(8a+1)x+a$
• $y^2=(6a+7)x^6+4ax^5+(5a+10)x^4+(9a+7)x^3+(10a+4)x^2+(7a+5)x+2a+1$
• $y^2=(7a+5)x^6+(4a+9)x^5+(9a+1)x^4+(2a+2)x^3+(7a+5)x^2+(5a+8)x+5a+10$
• $y^2=4x^6+(6a+1)x^5+(4a+4)x^4+(3a+4)x^3+(a+7)x^2+(9a+6)x+3a+2$
• $y^2=(7a+11)x^6+(11a+6)x^5+(11a+11)x^4+(a+6)x^3+(9a+10)x^2+12ax+3a+1$
• $y^2=(4a+8)x^6+(11a+12)x^5+(6a+7)x^4+(8a+2)x^3+(a+2)x^2+12ax+12a+1$
• $y^2=(9a+5)x^6+12ax^4+(2a+4)x^3+(8a+7)x^2+(7a+11)x+9a$
• $y^2=(9a+11)x^6+(9a+2)x^5+(8a+10)x^4+(4a+6)x^3+7x^2+(12a+6)x+3a+7$
• $y^2=(2a+7)x^6+(9a+2)x^5+(12a+5)x^4+(a+1)x^3+(4a+5)x^2+(3a+11)x+2a+7$
• and 28 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21890 806646500 23294046862130 665423815855370000 19004955701325285042450 542800612104003946591398500 15502932164795167161755753892770 442779262228407092408733446481920000 12646218550443873311595380911734303283010 361188648084055844278856303330219084420662500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 126 28242 4825974 815739558 137858433286 23298078329202 3937376223696174 665416606856166078 112455406931625121566 19004963774855774310802

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.14362880.1.
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_bfc $2$ (not in LMFDB)