Properties

Label 2.163.aby_bkp
Base field $\F_{163}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{163}$
Dimension:  $2$
L-polynomial:  $( 1 - 25 x + 163 x^{2} )^{2}$
  $1 - 50 x + 951 x^{2} - 8150 x^{3} + 26569 x^{4}$
Frobenius angles:  $\pm0.0652307277549$, $\pm0.0652307277549$
Angle rank:  $1$ (numerical)
Jacobians:  $2$

This isogeny class is not simple, 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $19321$ $690165441$ $18725940713104$ $498260220088499001$ $13239554876288343679561$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $114$ $25972$ $4323948$ $705839236$ $115062912294$ $18755363780998$ $3057125211164418$ $498311414511757828$ $81224760543592981524$ $13239635967230078250772$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{163}$.

Endomorphism algebra over $\F_{163}$
The isogeny class factors as 1.163.az 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.163.a_aln$2$(not in LMFDB)
2.163.by_bkp$2$(not in LMFDB)
2.163.ar_ew$3$(not in LMFDB)
2.163.ai_adv$3$(not in LMFDB)
2.163.q_pa$3$(not in LMFDB)
2.163.z_ru$3$(not in LMFDB)
2.163.bi_xr$3$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.163.a_aln$2$(not in LMFDB)
2.163.by_bkp$2$(not in LMFDB)
2.163.ar_ew$3$(not in LMFDB)
2.163.ai_adv$3$(not in LMFDB)
2.163.q_pa$3$(not in LMFDB)
2.163.z_ru$3$(not in LMFDB)
2.163.bi_xr$3$(not in LMFDB)
2.163.a_ln$4$(not in LMFDB)
2.163.abq_bcx$6$(not in LMFDB)
2.163.abi_xr$6$(not in LMFDB)
2.163.abh_ug$6$(not in LMFDB)
2.163.az_ru$6$(not in LMFDB)
2.163.aq_pa$6$(not in LMFDB)
2.163.aj_hi$6$(not in LMFDB)
2.163.a_bl$6$(not in LMFDB)
2.163.a_kc$6$(not in LMFDB)
2.163.i_adv$6$(not in LMFDB)
2.163.j_hi$6$(not in LMFDB)
2.163.r_ew$6$(not in LMFDB)
2.163.bh_ug$6$(not in LMFDB)
2.163.bq_bcx$6$(not in LMFDB)
2.163.a_akc$12$(not in LMFDB)
2.163.a_abl$12$(not in LMFDB)