Properties

Label 2.37.au_gq
Base field $\F_{37}$
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:  $\F_{37}$
Dimension:  $2$
L-polynomial:  $1 - 20 x + 172 x^{2} - 740 x^{3} + 1369 x^{4}$
Frobenius angles:  $\pm0.112452867781$, $\pm0.250611395750$
Angle rank:  $2$ (numerical)
Number field:  4.0.84224.1
Galois group:  $D_{4}$
Jacobians:  $4$
Isomorphism classes:  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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $782$ $1800164$ $2570822654$ $3516807591056$ $4809617773833582$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $18$ $1314$ $50754$ $1876470$ $69358858$ $2565778098$ $94931919114$ $3512479269534$ $129961746737778$ $4808584503290914$

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 $\F_{37}$.

Endomorphism algebra over $\F_{37}$
The endomorphism algebra of this simple isogeny class is 4.0.84224.1.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.37.u_gq$2$(not in LMFDB)