Properties

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

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Invariants

Base field:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )( 1 + 3 x^{2} )$
Frobenius angles:  $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.5$
Angle rank:  $1$ (numerical)
Jacobians:  0

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 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
$A(\F_{q^r})$ 8 1344 29792 559104 16002008 420424704 10435530344 277539225600 7700377178144 208429482617664

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 15 38 87 269 792 2183 6447 19874 59775

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 1.3.ac $\times$ 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.abu $\times$ 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.ab_d_ag$2$3.9.f_p_cc
3.3.b_d_g$2$3.9.f_p_cc
3.3.f_p_be$2$3.9.f_p_cc
3.3.ai_be_aco$3$(not in LMFDB)
3.3.ac_a_g$3$(not in LMFDB)
3.3.ac_j_am$3$(not in LMFDB)
3.3.b_d_g$3$(not in LMFDB)
3.3.e_g_g$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.ab_d_ag$2$3.9.f_p_cc
3.3.b_d_g$2$3.9.f_p_cc
3.3.f_p_be$2$3.9.f_p_cc
3.3.ai_be_aco$3$(not in LMFDB)
3.3.ac_a_g$3$(not in LMFDB)
3.3.ac_j_am$3$(not in LMFDB)
3.3.b_d_g$3$(not in LMFDB)
3.3.e_g_g$3$(not in LMFDB)
3.3.ae_g_ag$6$(not in LMFDB)
3.3.c_a_ag$6$(not in LMFDB)
3.3.c_j_m$6$(not in LMFDB)
3.3.i_be_co$6$(not in LMFDB)
3.3.ac_ad_m$12$(not in LMFDB)
3.3.ac_g_ag$12$(not in LMFDB)
3.3.c_ad_am$12$(not in LMFDB)
3.3.c_g_g$12$(not in LMFDB)
3.3.ac_d_a$24$(not in LMFDB)
3.3.c_d_a$24$(not in LMFDB)