Properties

Label 3.3.ag_r_abh
Base field $\F_{3}$
Dimension $3$
$p$-rank $2$
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:  $( 1 - 3 x + 3 x^{2} )( 1 - 3 x + 5 x^{2} - 9 x^{3} + 9 x^{4} )$
  $1 - 6 x + 17 x^{2} - 33 x^{3} + 51 x^{4} - 54 x^{5} + 27 x^{6}$
Frobenius angles:  $\pm0.0975263560046$, $\pm0.166666666667$, $\pm0.527857038681$
Angle rank:  $2$ (numerical)
Jacobians:  $0$
Isomorphism classes:  1

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $3$ $567$ $15372$ $449631$ $16923408$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-2$ $8$ $19$ $68$ $283$ $845$ $2266$ $6644$ $20143$ $59603$

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 $\times$ 2.3.ad_f 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.cj_ddt. 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.a_ab_d$2$3.9.ac_af_bz
3.3.g_r_bh$2$3.9.ac_af_bz
3.3.ad_i_as$3$(not in LMFDB)
3.3.a_ab_ad$3$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
3.3.a_ab_d$2$3.9.ac_af_bz
3.3.g_r_bh$2$3.9.ac_af_bz
3.3.ad_i_as$3$(not in LMFDB)
3.3.a_ab_ad$3$(not in LMFDB)
3.3.ad_i_as$6$(not in LMFDB)
3.3.d_i_s$6$(not in LMFDB)