# Properties

 Label 2.169.aby_blb 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: $( 1 - 25 x + 169 x^{2} )^{2}$ Frobenius angles: $\pm0.0885687144757$, $\pm0.0885687144757$ Angle rank: $1$ (numerical) Jacobians: 6

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 6 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21025 799475625 23269625299600 665375421944975625 19004927208209142600625 542800814692874264616960000 15502933166409639470199270448225 442779265099756037591146070376575625 12646218556563926064314771702828281728400 361188648094342857041324591376435597266015625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 120 27988 4820910 815680228 137858226600 23298087024718 3937376478082440 665416611171280708 112455406986047162430 19004963775397054624948

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The isogeny class factors as 1.169.az 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-51})$$$)$
All geometric endomorphisms are defined over $\F_{13^{2}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{13^{2}}$.

 Subfield Primitive Model $\F_{13}$ 2.13.a_az

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.169.a_alb $2$ (not in LMFDB) 2.169.by_blb $2$ (not in LMFDB) 2.169.z_ro $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.169.a_alb $2$ (not in LMFDB) 2.169.by_blb $2$ (not in LMFDB) 2.169.z_ro $3$ (not in LMFDB) 2.169.a_lb $4$ (not in LMFDB) 2.169.az_ro $6$ (not in LMFDB)