# Properties

 Label 2.169.abv_bid Base Field $\F_{13^{2}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

# Learn more about

## Invariants

 Base field: $\F_{13^{2}}$ Dimension: $2$ L-polynomial: $1 - 47 x + 887 x^{2} - 7943 x^{3} + 28561 x^{4}$ Frobenius angles: $\pm0.0738822392977$, $\pm0.185751622566$ Angle rank: $2$ (numerical) Number field: 4.0.1240629.1 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21459 803403501 23285614822227 665420910811195941 19005022762936426138224 542800941032059740838587429 15502933119425606590811877797979 442779264130864598597084153772781989 12646218552686199868611550903280740664307 361188648083380578164030039351771178021739776

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 123 28127 4824225 815735995 137858919738 23298092447447 3937376466149613 665416609715213299 112455406951564816935 19004963774820243271502

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.1240629.1.
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.bv_bid $2$ (not in LMFDB)