Properties

Label 1.331.as
Base Field $\F_{331}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{331}$
Dimension:  $1$
L-polynomial:  $1 - 18 x + 331 x^{2}$
Frobenius angles:  $\pm0.335283534355$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-10}) \)
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
$A(\F_{q^r})$ 314 109900 36276734 12003717600 3973193712554 1315127740845100 435307305600557774 144086718368760854400 47692703776190360242394 15786284949778201421447500

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 314 109900 36276734 12003717600 3973193712554 1315127740845100 435307305600557774 144086718368760854400 47692703776190360242394 15786284949778201421447500

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{331}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-10}) \).
All geometric endomorphisms are defined over $\F_{331}$.

Base change

This is a primitive isogeny class.

Twists

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