Properties

Label 1.397.ab
Base Field $\F_{397}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{397}$
Dimension:  $1$
L-polynomial:  $1 - x + 397 x^{2}$
Frobenius angles:  $\pm0.492011403795$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}) \)
Galois group:  $C_2$
Jacobians:  9

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 9 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 397 158403 62571964 24840283251 9861716175697 3915101757542976 1554295349061350677 617055253355663717763 244970935601304037979308 97253461433824820542554843

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 397 158403 62571964 24840283251 9861716175697 3915101757542976 1554295349061350677 617055253355663717763 244970935601304037979308 97253461433824820542554843

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{397}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \).
All geometric endomorphisms are defined over $\F_{397}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
1.397.b$2$(not in LMFDB)
1.397.abi$3$(not in LMFDB)
1.397.bj$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.397.b$2$(not in LMFDB)
1.397.abi$3$(not in LMFDB)
1.397.bj$3$(not in LMFDB)
1.397.abj$6$(not in LMFDB)
1.397.bi$6$(not in LMFDB)