Properties

Label 1.361.ai
Base field $\F_{19^{2}}$
Dimension $1$
$p$-rank $1$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{19^{2}}$
Dimension:  $1$
L-polynomial:  $1 - 8 x + 361 x^{2}$
Frobenius angles:  $\pm0.432482236283$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-345}) \)
Galois group:  $C_2$
Jacobians:  $8$

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $354$ $130980$ $47054034$ $16983390720$ $6131061936354$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $354$ $130980$ $47054034$ $16983390720$ $6131061936354$ $2213314946702820$ $799006687564020114$ $288441413571893422080$ $104127350297302429474914$ $37589973457539545413054500$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{19^{2}}$.

Endomorphism algebra over $\F_{19^{2}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-345}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
1.361.i$2$(not in LMFDB)