Properties

Label 2.9.ah_y
Base field $\F_{3^{2}}$
Dimension $2$
$p$-rank $1$
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:  $2$
L-polynomial:  $( 1 - 3 x )^{2}( 1 - x + 9 x^{2} )$
  $1 - 7 x + 24 x^{2} - 63 x^{3} + 81 x^{4}$
Frobenius angles:  $0$, $0$, $\pm0.446699620962$
Angle rank:  $1$ (numerical)
Jacobians:  $0$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $36$ $6336$ $511056$ $41184000$ $3437062596$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $3$ $81$ $702$ $6273$ $58203$ $530766$ $4782627$ $43030593$ $387341838$ $3486654081$

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 $\times$ 1.9.ab 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
2.9.af_m$2$2.81.ab_afo
2.9.f_m$2$2.81.ab_afo
2.9.h_y$2$2.81.ab_afo
2.9.c_p$3$2.729.abc_cc

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

TwistExtension degreeCommon base change
2.9.af_m$2$2.81.ab_afo
2.9.f_m$2$2.81.ab_afo
2.9.h_y$2$2.81.ab_afo
2.9.c_p$3$2.729.abc_cc
2.9.ab_s$4$(not in LMFDB)
2.9.b_s$4$(not in LMFDB)
2.9.ae_v$6$(not in LMFDB)
2.9.ac_p$6$(not in LMFDB)
2.9.e_v$6$(not in LMFDB)