# Properties

 Label 2.169.abu_bha Base Field $\F_{13^{2}}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{13^{2}}$ Dimension: $2$ L-polynomial: $( 1 - 13 x )^{2}( 1 - 20 x + 169 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.220639651288$ Angle rank: $1$ (numerical) Jacobians: 0

This isogeny class is not simple.

## Newton polygon $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1]$

## Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21600 804384000 23287205743200 665413472102400000 19004958441440194428000 542800663678457499391776000 15502932239712970008873815560800 442779261888025523712270923366400000 12646218547962601326904502478412935266400 361188648075114957512618021817280424973600000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 124 28162 4824556 815726878 137858453164 23298080542882 3937376242723516 665416606344634558 112455406909560624604 19004963774385324228802

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The isogeny class factors as 1.169.aba $\times$ 1.169.au and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.169.aba : the quaternion algebra over $$\Q$$ ramified at $13$ and $\infty$. 1.169.au : $$\Q(\sqrt{-69})$$.
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.ag_aha $2$ (not in LMFDB) 2.169.g_aha $2$ (not in LMFDB) 2.169.bu_bha $2$ (not in LMFDB)