Properties

Label 2.139.abq_bbq
Base field $\F_{139}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{139}$
Dimension:  $2$
L-polynomial:  $( 1 - 22 x + 139 x^{2} )( 1 - 20 x + 139 x^{2} )$
  $1 - 42 x + 718 x^{2} - 5838 x^{3} + 19321 x^{4}$
Frobenius angles:  $\pm0.117174211439$, $\pm0.177693164435$
Angle rank:  $2$ (numerical)
Jacobians:  $24$

This isogeny class is not 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $14160$ $367027200$ $7209508790160$ $139361120450150400$ $2692480707829065766800$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $98$ $18994$ $2684486$ $373321006$ $51889394018$ $7212557867362$ $1002544467555302$ $139353668126050846$ $19370159748397287074$ $2692452204206655038674$

Jacobians and polarizations

This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{139}$.

Endomorphism algebra over $\F_{139}$
The isogeny class factors as 1.139.aw $\times$ 1.139.au and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.139.ac_agg$2$(not in LMFDB)
2.139.c_agg$2$(not in LMFDB)
2.139.bq_bbq$2$(not in LMFDB)