Properties

Label 2.167.a_hi
Base field $\F_{167}$
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:  $\F_{167}$
Dimension:  $2$
L-polynomial:  $( 1 - 12 x + 167 x^{2} )( 1 + 12 x + 167 x^{2} )$
  $1 + 190x^{2} + 27889x^{4}$
Frobenius angles:  $\pm0.346308130027$, $\pm0.653691869973$
Angle rank:  $1$ (numerical)
Jacobians:  6375

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 6375 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $28080$ $788486400$ $21691952558640$ $604997729856000000$ $16871927924959158308400$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $168$ $28270$ $4657464$ $777835678$ $129891985608$ $21691943520910$ $3622557586593624$ $604967119297872958$ $101029508532509551848$ $16871927924989221458350$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{167}$
The isogeny class factors as 1.167.am $\times$ 1.167.m 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 $\overline{\F}_{167}$
The base change of $A$ to $\F_{167^{2}}$ is 1.27889.hi 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-131}) \)$)$
All geometric endomorphisms are defined over $\F_{167^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension degreeCommon base change
2.167.ay_sk$2$(not in LMFDB)
2.167.y_sk$2$(not in LMFDB)
2.167.a_ahi$4$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.167.ay_sk$2$(not in LMFDB)
2.167.y_sk$2$(not in LMFDB)
2.167.a_ahi$4$(not in LMFDB)
2.167.am_ax$6$(not in LMFDB)
2.167.m_ax$6$(not in LMFDB)

Additional information

This isogeny class contains the most Jacobians (6375) of any in the database.