Properties

Label 3.9.am_cs_ajy
Base field $\F_{3^{2}}$
Dimension $3$
$p$-rank $2$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{3^{2}}$
Dimension:  $3$
L-polynomial:  $( 1 - 3 x )^{2}( 1 - 6 x + 25 x^{2} - 54 x^{3} + 81 x^{4} )$
  $1 - 12 x + 70 x^{2} - 258 x^{3} + 630 x^{4} - 972 x^{5} + 729 x^{6}$
Frobenius angles:  $0$, $0$, $\pm0.236852280319$, $\pm0.414859841358$
Angle rank:  $2$ (numerical)

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

$p$-rank:  $2$
Slopes:  $[0, 0, 1/2, 1/2, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $188$ $502336$ $396641648$ $278947180800$ $203967451571708$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-2$ $78$ $748$ $6482$ $58498$ $530292$ $4780690$ $43028258$ $387318700$ $3486513438$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{2}}$.

Endomorphism algebra over $\F_{3^{2}}$
The isogeny class factors as 1.9.ag $\times$ 2.9.ag_z 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
3.9.a_ac_abq$2$(not in LMFDB)
3.9.a_ac_bq$2$(not in LMFDB)
3.9.m_cs_jy$2$(not in LMFDB)
3.9.ad_q_abh$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.9.a_ac_abq$2$(not in LMFDB)
3.9.a_ac_bq$2$(not in LMFDB)
3.9.m_cs_jy$2$(not in LMFDB)
3.9.ad_q_abh$3$(not in LMFDB)
3.9.ag_bi_aee$4$(not in LMFDB)
3.9.g_bi_ee$4$(not in LMFDB)
3.9.aj_ca_ahb$6$(not in LMFDB)
3.9.d_q_bh$6$(not in LMFDB)
3.9.j_ca_hb$6$(not in LMFDB)