Properties

Label 2.9.ah_be
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:  $( 1 - 4 x + 9 x^{2} )( 1 - 3 x + 9 x^{2} )$
Frobenius angles:  $\pm0.267720472801$, $\pm0.333333333333$
Angle rank:  $1$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

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

Point counts

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

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 42 7644 606816 44640960 3486431802 281402424576 22847046425322 1852792867511040 150105608160199776 12158088943167680604

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 3 93 828 6801 59043 529506 4776747 43041441 387448812 3486905853

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{2}}$
The isogeny class factors as 1.9.ae $\times$ 1.9.ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{3^{2}}$
The base change of $A$ to $\F_{3^{6}}$ is 1.729.bs $\times$ 1.729.cc. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3^{6}}$.

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.ab_g$2$2.81.l_gy
2.9.b_g$2$2.81.l_gy
2.9.h_be$2$2.81.l_gy
2.9.c_ag$3$2.729.du_frm
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.9.ab_g$2$2.81.l_gy
2.9.b_g$2$2.81.l_gy
2.9.h_be$2$2.81.l_gy
2.9.c_ag$3$2.729.du_frm
2.9.ak_bq$6$(not in LMFDB)
2.9.ac_ag$6$(not in LMFDB)
2.9.k_bq$6$(not in LMFDB)
2.9.ae_s$12$(not in LMFDB)
2.9.e_s$12$(not in LMFDB)