Properties

Label 3.3.ai_be_aco
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:  $( 1 - 2 x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}$
Frobenius angles:  $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.304086723985$
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
$A(\F_{q^r})$ 2 588 29792 794976 17772722 420424704 10821854822 288077043072 7700377178144 205877327739468

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -4 6 38 114 296 792 2264 6690 19874 59046

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 1.3.ac 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.ae_g_ag$2$3.9.ae_y_acc
3.3.ac_a_g$2$3.9.ae_y_acc
3.3.c_a_ag$2$3.9.ae_y_acc
3.3.e_g_g$2$3.9.ae_y_acc
3.3.i_be_co$2$3.9.ae_y_acc
3.3.af_p_abe$3$(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.ae_g_ag$2$3.9.ae_y_acc
3.3.ac_a_g$2$3.9.ae_y_acc
3.3.c_a_ag$2$3.9.ae_y_acc
3.3.e_g_g$2$3.9.ae_y_acc
3.3.i_be_co$2$3.9.ae_y_acc
3.3.af_p_abe$3$(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.ac_g_ag$4$(not in LMFDB)
3.3.c_g_g$4$(not in LMFDB)
3.3.ab_d_ag$6$(not in LMFDB)
3.3.c_j_m$6$(not in LMFDB)
3.3.f_p_be$6$(not in LMFDB)
3.3.ac_ad_m$12$(not in LMFDB)
3.3.c_ad_am$12$(not in LMFDB)
3.3.ac_d_a$24$(not in LMFDB)
3.3.c_d_a$24$(not in LMFDB)