Properties

Label 2.169.abu_bhj
Base field $\F_{13^{2}}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive no
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{13^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 23 x + 169 x^{2} )^{2}$
  $1 - 46 x + 867 x^{2} - 7774 x^{3} + 28561 x^{4}$
Frobenius angles:  $\pm0.154420958311$, $\pm0.154420958311$
Angle rank:  $1$ (numerical)
Jacobians:  $12$

This isogeny class is not simple, not primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $21609$ $804913641$ $23293210300416$ $665450286236357769$ $19005118247840541139449$

Point counts of the curve

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

Jacobians and polarizations

This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13^{2}}$.

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

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)