Properties

Label 2.193.abx_bls
Base field $\F_{193}$
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:  $\F_{193}$
Dimension:  $2$
L-polynomial:  $( 1 - 27 x + 193 x^{2} )( 1 - 22 x + 193 x^{2} )$
  $1 - 49 x + 980 x^{2} - 9457 x^{3} + 37249 x^{4}$
Frobenius angles:  $\pm0.0758389534121$, $\pm0.209145594264$
Angle rank:  $2$ (numerical)
Jacobians:  $27$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $28724$ $1371168864$ $51668455909184$ $1925153121045338496$ $71709076489180543513364$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $145$ $36809$ $7187098$ $1387509745$ $267785825065$ $51682548534878$ $9974730367878841$ $1925122952395687969$ $371548729898333227354$ $71708904873153875140889$

Jacobians and polarizations

This isogeny class contains the Jacobians of 27 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{193}$.

Endomorphism algebra over $\F_{193}$
The isogeny class factors as 1.193.abb $\times$ 1.193.aw 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
2.193.af_aia$2$(not in LMFDB)
2.193.f_aia$2$(not in LMFDB)
2.193.bx_bls$2$(not in LMFDB)