Properties

Label 2.131.abq_baz
Base field $\F_{131}$
Dimension $2$
$p$-rank $2$
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:  $\F_{131}$
Dimension:  $2$
L-polynomial:  $1 - 42 x + 701 x^{2} - 5502 x^{3} + 17161 x^{4}$
Frobenius angles:  $\pm0.0650836808263$, $\pm0.173182857662$
Angle rank:  $2$ (numerical)
Number field:  4.0.194112.5
Galois group:  $D_{4}$
Jacobians:  $4$

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:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $12319$ $288350833$ $5048818516516$ $86729045036858793$ $1488382950554161438519$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $90$ $16800$ $2245824$ $294495988$ $38579643330$ $5053916090238$ $662062655178966$ $86730203714621284$ $11361656654999045136$ $1488377021715468367440$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{131}$.

Endomorphism algebra over $\F_{131}$
The endomorphism algebra of this simple isogeny class is 4.0.194112.5.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.131.bq_baz$2$(not in LMFDB)