Properties

 Label 2.169.abx_bka 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: $( 1 - 13 x )^{2}( 1 - 23 x + 169 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.154420958311$ 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 $A(\F_{q^r})$ 21168 800743104 23274445630464 665386850963116800 19004938639139516899248 542800764409177865855287296 15502932786770818826863382108208 442779263493339365094039096398515200 12646218551154683096662435854401923574784 361188648078566017646134979180044812332915904

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 121 28033 4821910 815694241 137858309521 23298084866446 3937376381663209 665416608757128961 112455406937945936230 19004963774566911532753

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The isogeny class factors as 1.169.aba $\times$ 1.169.ax 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.ax : $$\Q(\sqrt{-3})$$.
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.ad_aka $2$ (not in LMFDB) 2.169.d_aka $2$ (not in LMFDB) 2.169.bx_bka $2$ (not in LMFDB) 2.169.az_ma $3$ (not in LMFDB) 2.169.ae_aja $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.169.ad_aka $2$ (not in LMFDB) 2.169.d_aka $2$ (not in LMFDB) 2.169.bx_bka $2$ (not in LMFDB) 2.169.az_ma $3$ (not in LMFDB) 2.169.ae_aja $3$ (not in LMFDB) 2.169.abw_bja $6$ (not in LMFDB) 2.169.abb_oa $6$ (not in LMFDB) 2.169.e_aja $6$ (not in LMFDB) 2.169.z_ma $6$ (not in LMFDB) 2.169.bb_oa $6$ (not in LMFDB) 2.169.bw_bja $6$ (not in LMFDB)