Properties

Label 2.139.abr_bcj
Base field $\F_{139}$
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:  $\F_{139}$
Dimension:  $2$
L-polynomial:  $1 - 43 x + 737 x^{2} - 5977 x^{3} + 19321 x^{4}$
Frobenius angles:  $\pm0.0488276070962$, $\pm0.185265935553$
Angle rank:  $2$ (numerical)
Number field:  4.0.368589.1
Galois group:  $D_{4}$
Jacobians:  $6$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $14039$ $366123081$ $7206201614537$ $139351783069244061$ $2692458192750605660624$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $97$ $18947$ $2683255$ $373295995$ $51888960112$ $7212551205287$ $1002544375960357$ $139353667002050179$ $19370159736341116855$ $2692452204100261958102$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 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 endomorphism algebra of this simple isogeny class is 4.0.368589.1.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.139.br_bcj$2$(not in LMFDB)