# Properties

 Label 2.169.abt_bgg 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: $( 1 - 25 x + 169 x^{2} )( 1 - 20 x + 169 x^{2} )$ Frobenius angles: $\pm0.0885687144757$, $\pm0.220639651288$ Angle rank: $2$ (numerical) Jacobians: 32

This isogeny class is not 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 32 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21750 805837500 23294178747000 665439475217400000 19005042530173093293750 542800910748553337462400000 15502932915715641778131850275750 442779263635084729775730687506400000 12646218552264457350883893894582674163000 361188648085260654355450131977349676148437500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 125 28213 4826000 815758753 137859063125 23298091147618 3937376414412125 665416608970146433 112455406947814508000 19004963774919168805573

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The isogeny class factors as 1.169.az $\times$ 1.169.au and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.af_agg $2$ (not in LMFDB) 2.169.f_agg $2$ (not in LMFDB) 2.169.bt_bgg $2$ (not in LMFDB)