Properties

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

Learn more about

Invariants

Base field:  $\F_{397}$
Dimension:  $1$
L-polynomial:  $1 - 16 x + 397 x^{2}$
Frobenius angles:  $\pm0.368486025368$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-37}) \)
Galois group:  $C_2$
Jacobians:  10

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 10 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 382 158148 62585734 24840622656 9861711435022 3915101535157476 1554295349241112342 617055253453363822848 244970935602065626123678 97253461433794893623636868

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 382 158148 62585734 24840622656 9861711435022 3915101535157476 1554295349241112342 617055253453363822848 244970935602065626123678 97253461433794893623636868

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{397}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-37}) \).
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.q$2$(not in LMFDB)