Properties

Label 2.139.abs_bdi
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:  $( 1 - 22 x + 139 x^{2} )^{2}$
  $1 - 44 x + 762 x^{2} - 6116 x^{3} + 19321 x^{4}$
Frobenius angles:  $\pm0.117174211439$, $\pm0.117174211439$
Angle rank:  $1$ (numerical)
Jacobians:  $11$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $13924$ $365421456$ $7204639749316$ $139350835364373504$ $2692464804794951989924$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $96$ $18910$ $2682672$ $373293454$ $51889087536$ $7212555810286$ $1002544475411904$ $139353668676098974$ $19370159759770597248$ $2692452204375011117950$

Jacobians and polarizations

This isogeny class contains the Jacobians of 11 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 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.139.a_ahy$2$(not in LMFDB)
2.139.bs_bdi$2$(not in LMFDB)
2.139.w_nh$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.139.a_ahy$2$(not in LMFDB)
2.139.bs_bdi$2$(not in LMFDB)
2.139.w_nh$3$(not in LMFDB)
2.139.a_hy$4$(not in LMFDB)
2.139.aw_nh$6$(not in LMFDB)
2.139.am_cu$8$(not in LMFDB)
2.139.m_cu$8$(not in LMFDB)