Properties

Label 1.397.ae
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:  $1 - 4 x + 397 x^{2}$
Frobenius angles:  $\pm0.467995070655$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-393}) \)
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
$A(\F_{q^r})$ 394 158388 62575474 24840306816 9861713935594 3915101736869076 1554295350239152306 617055253369889837568 244970935600747305580618 97253461433816280765895668

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 394 158388 62575474 24840306816 9861713935594 3915101736869076 1554295350239152306 617055253369889837568 244970935600747305580618 97253461433816280765895668

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{397}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-393}) \).
All geometric endomorphisms are defined over $\F_{397}$.

Base change

This is a primitive isogeny class.

Twists

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