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

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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:

Point counts of the abelian variety

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

Point counts of the curve

$r$ 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 $\F_{13^{2}}$.

SubfieldPrimitive 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.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)