Properties

Label 1.331.ax
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:  $1 - 23 x + 331 x^{2}$
Frobenius angles:  $\pm0.282194346064$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-795}) \)
Galois group:  $C_2$
Jacobians:  4

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 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 309 109695 36275364 12003814155 3973196911179 1315127771963280 435307304895129129 144086718339184995795 47692703775808651528764 15786284949781392276987375

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 309 109695 36275364 12003814155 3973196911179 1315127771963280 435307304895129129 144086718339184995795 47692703775808651528764 15786284949781392276987375

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{331}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-795}) \).
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.x$2$(not in LMFDB)