Properties

Label 3.3.af_m_av
Base field $\F_{3}$
Dimension $3$
$p$-rank $1$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

Related objects

Downloads

Learn more

Invariants

Base field:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $( 1 + x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}$
  $1 - 5 x + 12 x^{2} - 21 x^{3} + 36 x^{4} - 45 x^{5} + 27 x^{6}$
Frobenius angles:  $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.593214749339$
Angle rank:  $1$ (numerical)
Jacobians:  $0$

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $5$ $735$ $15680$ $621075$ $20196275$

Point counts of the (virtual) curve

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

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{6}}$.

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:
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)