Properties

Label 2.169.abv_bif
Base field $\F_{13^{2}}$
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:  $\F_{13^{2}}$
Dimension:  $2$
L-polynomial:  $1 - 47 x + 889 x^{2} - 7943 x^{3} + 28561 x^{4}$
Frobenius angles:  $\pm0.104248499198$, $\pm0.169935694661$
Angle rank:  $2$ (numerical)
Number field:  4.0.306125.1
Galois group:  $D_{4}$
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:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $21461$ $803521301$ $23286978302621$ $665429532354810245$ $19005061768865309085696$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $123$ $28131$ $4824507$ $815746563$ $137859202678$ $23298098465811$ $3937376572510707$ $665416611294768963$ $112455406970810835843$ $19004963774992452783406$

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 $\F_{13^{2}}$.

Endomorphism algebra over $\F_{13^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.306125.1.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.169.bv_bif$2$(not in LMFDB)