# Properties

 Label 2.169.abu_bhc 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: $1 - 46 x + 860 x^{2} - 7774 x^{3} + 28561 x^{4}$ Frobenius angles: $\pm0.0526049868893$, $\pm0.213762271581$ Angle rank: $2$ (numerical) Number field: 4.0.417088.1 Galois group: $D_{4}$ Jacobians: 20

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 20 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21602 804501684 23288540053058 665421675820774992 19004994397849796299922 542800789223188122956132244 15502932609596279221187336417426 442779262844000626920624729791321088 12646218550214444006776172026231311273746 361188648080201982004447667018931063900038484

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 124 28166 4824832 815736934 137858713984 23298085931510 3937376336665084 665416607781291070 112455406929584941276 19004963774652992431286

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.417088.1.
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.bu_bhc $2$ (not in LMFDB)