Properties

Label 1.397.a
Base Field $\F_{397}$
Dimension $1$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{397}$
Dimension:  $1$
L-polynomial:  $1 + 397 x^{2}$
Frobenius angles:  $\pm0.5$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-397}) \)
Galois group:  $C_2$
Jacobians:  6

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2]$

Point counts

This isogeny class contains the Jacobians of 6 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 398 158404 62570774 24840281664 9861716961758 3915101758959076 1554295348625559014 617055253354665734400 244970935601525730479918 97253461433825438434450564

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 398 158404 62570774 24840281664 9861716961758 3915101758959076 1554295348625559014 617055253354665734400 244970935601525730479918 97253461433825438434450564

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{397}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-397}) \).
Endomorphism algebra over $\overline{\F}_{397}$
The base change of $A$ to $\F_{397^{2}}$ is the simple isogeny class 1.157609.beo and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $397$ and $\infty$.
All geometric endomorphisms are defined over $\F_{397^{2}}$.

Base change

This is a primitive isogeny class.

Twists

This isogeny class has no twists.