Properties

Label 2.17.am_co
Base field $\F_{17}$
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:  $\F_{17}$
Dimension:  $2$
L-polynomial:  $( 1 - 8 x + 17 x^{2} )( 1 - 4 x + 17 x^{2} )$
  $1 - 12 x + 66 x^{2} - 204 x^{3} + 289 x^{4}$
Frobenius angles:  $\pm0.0779791303774$, $\pm0.338793663197$
Angle rank:  $2$ (numerical)
Jacobians:  $4$
Isomorphism classes:  12

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $140$ $80080$ $24309740$ $6970163200$ $2012913910700$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $6$ $278$ $4950$ $83454$ $1417686$ $24126806$ $410328582$ $6975923326$ $118588965030$ $2015997066518$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{17}$.

Endomorphism algebra over $\F_{17}$
The isogeny class factors as 1.17.ai $\times$ 1.17.ae 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 are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.17.ae_c$2$(not in LMFDB)
2.17.e_c$2$(not in LMFDB)
2.17.m_co$2$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.17.ae_c$2$(not in LMFDB)
2.17.e_c$2$(not in LMFDB)
2.17.m_co$2$(not in LMFDB)
2.17.ag_bq$4$(not in LMFDB)
2.17.ac_ba$4$(not in LMFDB)
2.17.c_ba$4$(not in LMFDB)
2.17.g_bq$4$(not in LMFDB)