# Properties

 Label 2.169.aby_bla 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: $( 1 - 13 x )^{2}( 1 - 24 x + 169 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.125665916378$ 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 $A(\F_{q^r})$ 21024 799416576 23268899852064 665370404536320000 19004901566295727075104 542800706483172827618724096 15502932767769163053791360922144 442779263776688659925320126955520000 12646218552531220140261393123616485425184 361188648082918367988225092855510083772610816

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 120 27986 4820760 815674078 137858040600 23298082380146 3937376376837240 665416609182951358 112455406950186673080 19004963774795922773906

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The isogeny class factors as 1.169.aba $\times$ 1.169.ay 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.ay : $$\Q(\sqrt{-1})$$.
All geometric endomorphisms are defined over $\F_{13^{2}}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.169.ac_ala $2$ (not in LMFDB) 2.169.c_ala $2$ (not in LMFDB) 2.169.by_bla $2$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.169.ac_ala $2$ (not in LMFDB) 2.169.c_ala $2$ (not in LMFDB) 2.169.by_bla $2$ (not in LMFDB) 2.169.abk_xa $4$ (not in LMFDB) 2.169.aq_da $4$ (not in LMFDB) 2.169.q_da $4$ (not in LMFDB) 2.169.bk_xa $4$ (not in LMFDB)