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

Learn more about

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.

Point counts of the abelian variety

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

Point counts of the (virtual) curve

$r$ 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

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)