# Properties

 Label 2.169.abs_bfa Base Field $\F_{13^{2}}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{13^{2}}$ Dimension: $2$ L-polynomial: $( 1 - 13 x )^{2}( 1 - 18 x + 169 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.256594102998$ Angle rank: $1$ (numerical) Jacobians: 12

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1]$

## Point counts

This isogeny class contains the Jacobians of 12 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21888 806529024 23292770811264 665416447679692800 19004925918315703563648 542800517580047118522525696 15502931912308550212098714123648 442779261620511949717910235394867200 12646218549000629892461926414396419609984 361188648080369222933585830417600853326178304

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 126 28238 4825710 815730526 137858217246 23298074272046 3937376159570574 665416605942610366 112455406918791204030 19004963774661792286478

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The isogeny class factors as 1.169.aba $\times$ 1.169.as and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.169.aba : the quaternion algebra over $$\Q$$ ramified at $13$ and $\infty$. 1.169.as : $$\Q(\sqrt{-22})$$.
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.ai_afa $2$ (not in LMFDB) 2.169.i_afa $2$ (not in LMFDB) 2.169.bs_bfa $2$ (not in LMFDB)