# Properties

 Label 2.169.abt_bgh 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: $1 - 45 x + 839 x^{2} - 7605 x^{3} + 28561 x^{4}$ Frobenius angles: $\pm0.0974399185523$, $\pm0.216609910606$ Angle rank: $2$ (numerical) Number field: 4.0.7245189.1 Galois group: $D_{4}$ Jacobians: 12

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 12 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21751 805896301 23294831363239 665443347009001029 19005058566634093568176 542800961870456101271317621 15502933044905016862399804525951 442779263887007509261828388940321189 12646218552584336947644950532727540576759 361188648085211868841053499997619840786846976

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 125 28215 4826135 815763499 137859179450 23298093341871 3937376447223155 665416609348740499 112455406950659009705 19004963774916601817550

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.7245189.1.
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.bt_bgh $2$ (not in LMFDB)