Properties

Label 1.331.abg
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:  $1 - 32 x + 331 x^{2}$
Frobenius angles:  $\pm0.157917618260$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}) \)
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
$A(\F_{q^r})$ 300 109200 36263700 12003700800 3973198957500 1315127884870800 435307307458577700 144086718372002515200 47692703775861198420300 15786284949772700784330000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 300 109200 36263700 12003700800 3973198957500 1315127884870800 435307307458577700 144086718372002515200 47692703775861198420300 15786284949772700784330000

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{331}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \).
All geometric endomorphisms are defined over $\F_{331}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
1.331.bg$2$(not in LMFDB)
1.331.b$3$(not in LMFDB)
1.331.bf$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.331.bg$2$(not in LMFDB)
1.331.b$3$(not in LMFDB)
1.331.bf$3$(not in LMFDB)
1.331.abf$6$(not in LMFDB)
1.331.ab$6$(not in LMFDB)