# Properties

 Label 2.169.abu_bhj 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: $( 1 - 23 x + 169 x^{2} )^{2}$ Frobenius angles: $\pm0.154420958311$, $\pm0.154420958311$ Angle rank: $1$ (numerical) Jacobians: 12

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 12 curves, and hence is principally polarizable:

• $y^2=8ax^6+9ax^5+7ax^4+10ax^3+7ax^2+6ax+11a$
• $y^2=6ax^6+11ax^5+4ax^4+4ax^3+4ax^2+4ax+10a$
• $y^2=(4a+4)x^6+(4a+4)x^5+(4a+2)x^4+(12a+1)x^3+(5a+11)x^2+(6a+11)x+9a+8$
• $y^2=(9a+8)x^6+(9a+8)x^5+(9a+6)x^4+ax^3+(8a+3)x^2+(7a+4)x+4a+4$
• $y^2=x^6+6x^3+8$
• $y^2=(9a+12)x^6+(8a+3)x^5+(12a+10)x^4+(a+9)x^3+(7a+7)x^2+(10a+3)x+8a$
• $y^2=(12a+9)x^6+(a+7)x^5+(a+6)x^4+(11a+4)x^3+(8a+9)x^2+(a+10)x+10a+11$
• $y^2=(5a+9)x^6+(3a+12)x^5+(6a+9)x^4+(6a+7)x^3+(9a+3)x^2+(10a+8)x+10a+12$
• $y^2=(7a+3)x^6+(4a+3)x^5+(10a+2)x^4+(8a+12)x^3+(4a+5)x^2+(4a+12)x+12a$
• $y^2=4ax^6+(11a+2)x^5+(7a+10)x^4+(11a+5)x^3+(8a+2)x^2+(a+10)x+5a+9$
• $y^2=(6a+10)x^6+(9a+7)x^5+(3a+12)x^4+(5a+7)x^3+(9a+9)x^2+(9a+3)x+a+12$
• $y^2=4ax^6+(12a+11)x^5+(12a+3)x^4+(7a+8)x^3+(9a+11)x^2+(6a+3)x+3a+4$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21609 804913641 23293210300416 665450286236357769 19005118247840541139449 542801208265836186498564096 15502933759137395096248079819049 442779265381041117436590161303702409 12646218554349152179237549800814788120576 361188648083080571936891258908884413177635401

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 124 28180 4825798 815772004 137859612364 23298103917646 3937376628621196 665416611594000964 112455406966352476822 19004963774804457594100

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The isogeny class factors as 1.169.ax 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
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.ao_cx $\F_{13}$ 2.13.a_ax $\F_{13}$ 2.13.o_cx

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.169.a_ahj $2$ (not in LMFDB) 2.169.bu_bhj $2$ (not in LMFDB) 2.169.aw_md $3$ (not in LMFDB) 2.169.ab_agm $3$ (not in LMFDB) 2.169.c_nb $3$ (not in LMFDB) 2.169.x_nw $3$ (not in LMFDB) 2.169.bs_bfq $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.169.a_ahj $2$ (not in LMFDB) 2.169.bu_bhj $2$ (not in LMFDB) 2.169.aw_md $3$ (not in LMFDB) 2.169.ab_agm $3$ (not in LMFDB) 2.169.c_nb $3$ (not in LMFDB) 2.169.x_nw $3$ (not in LMFDB) 2.169.bs_bfq $3$ (not in LMFDB) 2.169.a_hj $4$ (not in LMFDB) 2.169.abt_bgm $6$ (not in LMFDB) 2.169.abs_bfq $6$ (not in LMFDB) 2.169.ay_nx $6$ (not in LMFDB) 2.169.ax_nw $6$ (not in LMFDB) 2.169.av_me $6$ (not in LMFDB) 2.169.ac_nb $6$ (not in LMFDB) 2.169.a_afq $6$ (not in LMFDB) 2.169.a_mz $6$ (not in LMFDB) 2.169.b_agm $6$ (not in LMFDB) 2.169.v_me $6$ (not in LMFDB) 2.169.w_md $6$ (not in LMFDB) 2.169.y_nx $6$ (not in LMFDB) 2.169.bt_bgm $6$ (not in LMFDB) 2.169.a_amz $12$ (not in LMFDB) 2.169.a_fq $12$ (not in LMFDB)