Properties

Label 1.167.a
Base Field $\F_{167}$
Dimension $1$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{167}$
Dimension:  $1$
L-polynomial:  $1 + 167 x^{2}$
Frobenius angles:  $\pm0.5$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-167}) \)
Galois group:  $C_2$
Jacobians:  22

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2]$

Point counts

This isogeny class contains the Jacobians of 22 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 168 28224 4657464 777740544 129891985608 21691970911296 3622557586593624 604967115405542400 101029508532509551848 16871927925188879129664

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 168 28224 4657464 777740544 129891985608 21691970911296 3622557586593624 604967115405542400 101029508532509551848 16871927925188879129664

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{167}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-167}) \).
Endomorphism algebra over $\overline{\F}_{167}$
The base change of $A$ to $\F_{167^{2}}$ is the simple isogeny class 1.27889.mw and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $167$ and $\infty$.
All geometric endomorphisms are defined over $\F_{167^{2}}$.

Base change

This is a primitive isogeny class.

Twists

This isogeny class has no twists.