Properties

Label 1.343.aba
Base Field $\F_{7^{3}}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{7^{3}}$
Dimension:  $1$
L-polynomial:  $1 - 26 x + 343 x^{2}$
Frobenius angles:  $\pm0.252320125077$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-174}) \)
Galois group:  $C_2$
Jacobians:  12

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 12 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 318 117660 40362786 13841522400 4747564477038 1628413594381980 558545862973830546 191581231352930889600 65712362363196299032158 22539340290692949564078300

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 318 117660 40362786 13841522400 4747564477038 1628413594381980 558545862973830546 191581231352930889600 65712362363196299032158 22539340290692949564078300

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{3}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-174}) \).
All geometric endomorphisms are defined over $\F_{7^{3}}$.

Base change

This is a primitive isogeny class.

Twists

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