# Properties

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

## Invariants

 Base field: $\F_{13^{2}}$ Dimension: $2$ L-polynomial: $( 1 - 23 x + 169 x^{2} )( 1 - 22 x + 169 x^{2} )$ Frobenius angles: $\pm0.154420958311$, $\pm0.178912375022$ Angle rank: $1$ (numerical) Jacobians: 0

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 is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21756 806190336 23298094520064 665462657105777664 19005137818449437301276 542801208265836186498564096 15502933627672366335063290734236 442779264811432019055428938322694144 12646218552730347184442595286240450109184 361188648079674460262151771506378909506258176

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 125 28225 4826810 815787169 137859754325 23298103917646 3937376595232205 665416610737982209 112455406951957393130 19004963774625235380625

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The isogeny class factors as 1.169.ax $\times$ 1.169.aw and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{13^{2}}$
 The base change of $A$ to $\F_{13^{12}}$ is 1.23298085122481.uortm 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
All geometric endomorphisms are defined over $\F_{13^{12}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{13^{4}}$  The base change of $A$ to $\F_{13^{4}}$ is 1.28561.ahj $\times$ 1.28561.afq. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{13^{6}}$  The base change of $A$ to $\F_{13^{6}}$ is 1.4826809.atm $\times$ 1.4826809.tm. The endomorphism algebra for each factor is:

## 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.ab_agm $2$ (not in LMFDB) 2.169.b_agm $2$ (not in LMFDB) 2.169.bt_bgm $2$ (not in LMFDB) 2.169.ay_nx $3$ (not in LMFDB) 2.169.av_me $3$ (not in LMFDB) 2.169.a_ahj $3$ (not in LMFDB) 2.169.a_afq $3$ (not in LMFDB) 2.169.a_mz $3$ (not in LMFDB) 2.169.v_me $3$ (not in LMFDB) 2.169.y_nx $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.169.ab_agm $2$ (not in LMFDB) 2.169.b_agm $2$ (not in LMFDB) 2.169.bt_bgm $2$ (not in LMFDB) 2.169.ay_nx $3$ (not in LMFDB) 2.169.av_me $3$ (not in LMFDB) 2.169.a_ahj $3$ (not in LMFDB) 2.169.a_afq $3$ (not in LMFDB) 2.169.a_mz $3$ (not in LMFDB) 2.169.v_me $3$ (not in LMFDB) 2.169.y_nx $3$ (not in LMFDB) 2.169.abu_bhj $6$ (not in LMFDB) 2.169.abs_bfq $6$ (not in LMFDB) 2.169.ax_nw $6$ (not in LMFDB) 2.169.aw_md $6$ (not in LMFDB) 2.169.ac_nb $6$ (not in LMFDB) 2.169.b_agm $6$ (not in LMFDB) 2.169.c_nb $6$ (not in LMFDB) 2.169.w_md $6$ (not in LMFDB) 2.169.x_nw $6$ (not in LMFDB) 2.169.bs_bfq $6$ (not in LMFDB) 2.169.bu_bhj $6$ (not in LMFDB) 2.169.a_amz $12$ (not in LMFDB) 2.169.a_fq $12$ (not in LMFDB) 2.169.a_hj $12$ (not in LMFDB)