Properties

Label 1.397.abm
Base field $\F_{397}$
Dimension $1$
$p$-rank $1$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{397}$
Dimension:  $1$
L-polynomial:  $1 - 38 x + 397 x^{2}$
Frobenius angles:  $\pm0.0973642687429$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-1}) \)
Galois group:  $C_2$
Jacobians:  $8$

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $360$ $156960$ $62561160$ $24840489600$ $9861716701800$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $360$ $156960$ $62561160$ $24840489600$ $9861716701800$ $3915101666530080$ $1554295349971839240$ $617055253442518694400$ $244970935602441784347240$ $97253461433825370856288800$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{397}$.

Endomorphism algebra over $\F_{397}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1}) \).

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.bm$2$(not in LMFDB)
1.397.am$4$(not in LMFDB)
1.397.m$4$(not in LMFDB)