Properties

Label 3.3.ae_m_ay
Base Field $\F_{3}$
Dimension $3$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 - x + 3 x^{2} )( 1 + 3 x^{2} )$
Frobenius angles:  $\pm0.166666666667$, $\pm0.406785250661$, $\pm0.5$
Angle rank:  $1$ (numerical)
Jacobians:  1

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 12 1680 28224 436800 14084412 442552320 11274543012 283788960000 7574065265088 205078897808400

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 18 36 66 240 828 2352 6594 19548 58818

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 1.3.ab $\times$ 1.3.a 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 $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{6}}$ is 1.729.ak $\times$ 1.729.cc 2 . The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Remainder of endomorphism lattice by field

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.ac_g_am$2$3.9.i_y_cc
3.3.c_g_m$2$3.9.i_y_cc
3.3.e_m_y$2$3.9.i_y_cc
3.3.ah_y_abz$3$(not in LMFDB)
3.3.ab_a_d$3$(not in LMFDB)
3.3.ab_j_ag$3$(not in LMFDB)
3.3.c_g_m$3$(not in LMFDB)
3.3.f_m_v$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.ac_g_am$2$3.9.i_y_cc
3.3.c_g_m$2$3.9.i_y_cc
3.3.e_m_y$2$3.9.i_y_cc
3.3.ah_y_abz$3$(not in LMFDB)
3.3.ab_a_d$3$(not in LMFDB)
3.3.ab_j_ag$3$(not in LMFDB)
3.3.c_g_m$3$(not in LMFDB)
3.3.f_m_v$3$(not in LMFDB)
3.3.af_m_av$6$(not in LMFDB)
3.3.b_a_ad$6$(not in LMFDB)
3.3.b_j_g$6$(not in LMFDB)
3.3.h_y_bz$6$(not in LMFDB)
3.3.ab_ad_g$12$(not in LMFDB)
3.3.ab_g_ad$12$(not in LMFDB)
3.3.b_ad_ag$12$(not in LMFDB)
3.3.b_g_d$12$(not in LMFDB)
3.3.ab_d_a$24$(not in LMFDB)
3.3.b_d_a$24$(not in LMFDB)