Properties

Label 3.3.ag_t_abn
Base Field $\F_{3}$
Dimension $3$
Ordinary No
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian No

Learn more about

Invariants

Base field:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 - 3 x + 7 x^{2} - 9 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.166666666667$, $\pm0.227267020856$, $\pm0.464830336654$
Angle rank:  $2$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

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

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
$A(\F_{q^r})$ 5 1015 29540 593775 16802000 455743120 11052196895 275721202575 7429279071620 204702932896000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 12 37 92 283 849 2308 6404 19171 58707

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 2.3.ad_h 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.cc $\times$ 2.729.cn_dov. 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.a_b_ad$2$3.9.c_h_bz
3.3.a_b_d$2$3.9.c_h_bz
3.3.g_t_bn$2$3.9.c_h_bz
3.3.ad_k_as$3$(not in LMFDB)
3.3.a_b_d$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.a_b_ad$2$3.9.c_h_bz
3.3.a_b_d$2$3.9.c_h_bz
3.3.g_t_bn$2$3.9.c_h_bz
3.3.ad_k_as$3$(not in LMFDB)
3.3.a_b_d$3$(not in LMFDB)
3.3.d_k_s$6$(not in LMFDB)