Properties

Label 1.361.al
Base Field $\F_{19^{2}}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{19^{2}}$
Dimension:  $1$
L-polynomial:  $1 - 11 x + 361 x^{2}$
Frobenius angles:  $\pm0.406519728374$
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
$A(\F_{q^r})$ 351 130923 47056464 16983462483 6131061331551 2213314901179200 799006687364504151 288441413591476181283 104127350297602681851984 37589973457533952376956203

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 351 130923 47056464 16983462483 6131061331551 2213314901179200 799006687364504151 288441413591476181283 104127350297602681851984 37589973457533952376956203

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19^{2}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \).
All geometric endomorphisms are defined over $\F_{19^{2}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{19^{2}}$.

SubfieldPrimitive Model
$\F_{19}$1.19.ah
$\F_{19}$1.19.h

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
1.361.l$2$(not in LMFDB)
1.361.aba$3$(not in LMFDB)
1.361.bl$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.361.l$2$(not in LMFDB)
1.361.aba$3$(not in LMFDB)
1.361.bl$3$(not in LMFDB)
1.361.abl$6$(not in LMFDB)
1.361.ba$6$(not in LMFDB)