Properties

Label 1.179.af
Base Field $\F_{179}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{179}$
Dimension:  $1$
L-polynomial:  $1 - 5 x + 179 x^{2}$
Frobenius angles:  $\pm0.440169320799$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-691}) \)
Galois group:  $C_2$
Jacobians:  5

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

This isogeny class contains the Jacobians of 5 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 175 32375 5737900 1026578875 183765304625 32894118362000 5888046455143475 1053960288751069875 188658891683809575700 33769941616171564934375

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 175 32375 5737900 1026578875 183765304625 32894118362000 5888046455143475 1053960288751069875 188658891683809575700 33769941616171564934375

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{179}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-691}) \).
All geometric endomorphisms are defined over $\F_{179}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.179.f$2$(not in LMFDB)