# Properties

 Label 2.169.abw_bjd 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: $( 1 - 25 x + 169 x^{2} )( 1 - 23 x + 169 x^{2} )$ Frobenius angles: $\pm0.0885687144757$, $\pm0.154420958311$ Angle rank: $2$ (numerical) Jacobians: 14

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 14 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21315 802190025 23281414813440 665412853037812425 19005022727784799137075 542801011479319554180710400 15502933462773514450484042880195 442779265240398577491531563986050825 12646218555456539121727675671671789633280 361188648088711714489064028737538772038875625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 122 28084 4823354 815726116 137858919482 23298095471182 3937376553351818 665416611382640836 112455406976199819626 19004963775100756109524

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The isogeny class factors as 1.169.az $\times$ 1.169.ax 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 are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.169.ac_ajd $2$ (not in LMFDB) 2.169.c_ajd $2$ (not in LMFDB) 2.169.bw_bjd $2$ (not in LMFDB) 2.169.ay_mb $3$ (not in LMFDB) 2.169.ad_aie $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.169.ac_ajd $2$ (not in LMFDB) 2.169.c_ajd $2$ (not in LMFDB) 2.169.bw_bjd $2$ (not in LMFDB) 2.169.ay_mb $3$ (not in LMFDB) 2.169.ad_aie $3$ (not in LMFDB) 2.169.abv_bie $6$ (not in LMFDB) 2.169.aba_nz $6$ (not in LMFDB) 2.169.d_aie $6$ (not in LMFDB) 2.169.y_mb $6$ (not in LMFDB) 2.169.ba_nz $6$ (not in LMFDB) 2.169.bv_bie $6$ (not in LMFDB)