# Properties

 Label 2.169.abw_bjc 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: $1 - 48 x + 912 x^{2} - 8112 x^{3} + 28561 x^{4}$ Frobenius angles: $\pm0.0676966056373$, $\pm0.164966217013$ Angle rank: $2$ (numerical) Number field: 4.0.319744.3 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21314 802131076 23280718508642 665408312165921296 19005001287878055651074 542800929833940016331228164 15502933199020570246714861216034 442779264500695321507682933532524544 12646218553647734868744610322313327293762 361188648084922927460690755824233161828447876

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 122 28082 4823210 815720550 137858763962 23298091966802 3937376486364842 665416610271001534 112455406960115183930 19004963774901398348402

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.319744.3.
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.bw_bjc $2$ (not in LMFDB)