Properties

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

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Invariants

Base field:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $( 1 - x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )^{2}$
Frobenius angles:  $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.406785250661$
Angle rank:  $2$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

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
$A(\F_{q^r})$ 12 2160 51984 691200 12474132 336856320 10034195484 284453683200 7720988424048 207593257330800

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 19 56 103 209 628 2099 6607 19928 59539

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ac 2 $\times$ 1.3.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3}$.

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.ad_j_ao$2$3.9.j_bz_ha
3.3.ab_f_ac$2$3.9.j_bz_ha
3.3.b_f_c$2$3.9.j_bz_ha
3.3.d_j_o$2$3.9.j_bz_ha
3.3.f_r_bi$2$3.9.j_bz_ha
3.3.b_c_l$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.ad_j_ao$2$3.9.j_bz_ha
3.3.ab_f_ac$2$3.9.j_bz_ha
3.3.b_f_c$2$3.9.j_bz_ha
3.3.d_j_o$2$3.9.j_bz_ha
3.3.f_r_bi$2$3.9.j_bz_ha
3.3.b_c_l$3$(not in LMFDB)
3.3.ab_b_c$4$(not in LMFDB)
3.3.b_b_ac$4$(not in LMFDB)
3.3.ad_g_an$6$(not in LMFDB)
3.3.ab_c_al$6$(not in LMFDB)
3.3.d_g_n$6$(not in LMFDB)
3.3.af_p_abg$8$(not in LMFDB)
3.3.ad_h_aq$8$(not in LMFDB)
3.3.d_h_q$8$(not in LMFDB)
3.3.f_p_bg$8$(not in LMFDB)