Properties

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

Learn more about

Invariants

Base field:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $( 1 + x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}$
Frobenius angles:  $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.593214749339$
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
$A(\F_{q^r})$ 5 735 15680 621075 20196275 442552320 10837299905 294564072075 7679454345920 202567769810175

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 9 20 93 329 828 2267 6837 19820 58089

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 1.3.b 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.ah_y_abz$2$3.9.ab_g_bb
3.3.ab_a_d$2$3.9.ab_g_bb
3.3.b_a_ad$2$3.9.ab_g_bb
3.3.f_m_v$2$3.9.ab_g_bb
3.3.h_y_bz$2$3.9.ab_g_bb
3.3.ac_g_am$3$(not in LMFDB)
3.3.b_a_ad$3$(not in LMFDB)
3.3.b_j_g$3$(not in LMFDB)
3.3.e_m_y$3$(not in LMFDB)
3.3.h_y_bz$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.ah_y_abz$2$3.9.ab_g_bb
3.3.ab_a_d$2$3.9.ab_g_bb
3.3.b_a_ad$2$3.9.ab_g_bb
3.3.f_m_v$2$3.9.ab_g_bb
3.3.h_y_bz$2$3.9.ab_g_bb
3.3.ac_g_am$3$(not in LMFDB)
3.3.b_a_ad$3$(not in LMFDB)
3.3.b_j_g$3$(not in LMFDB)
3.3.e_m_y$3$(not in LMFDB)
3.3.h_y_bz$3$(not in LMFDB)
3.3.ab_g_ad$4$(not in LMFDB)
3.3.b_g_d$4$(not in LMFDB)
3.3.ah_y_abz$6$(not in LMFDB)
3.3.ae_m_ay$6$(not in LMFDB)
3.3.ac_g_am$6$(not in LMFDB)
3.3.ab_a_d$6$(not in LMFDB)
3.3.ab_j_ag$6$(not in LMFDB)
3.3.b_a_ad$6$(not in LMFDB)
3.3.b_j_g$6$(not in LMFDB)
3.3.c_g_m$6$(not in LMFDB)
3.3.e_m_y$6$(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)