Properties

Label 3.3.aj_bk_add
Base Field $\F_{3}$
Dimension $3$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

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

This isogeny class is not simple.

Newton polygon

This isogeny class is supersingular.

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

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
$A(\F_{q^r})$ 1 343 21952 753571 19902511 481890304 11681631109 293151929707 7626759805504 203370086883943

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -5 1 28 109 325 892 2431 6805 19684 58321

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
Endomorphism algebra over $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
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.ad_a_j$2$3.9.aj_cc_ahh
3.3.d_a_aj$2$3.9.aj_cc_ahh
3.3.j_bk_dd$2$3.9.aj_cc_ahh
3.3.ag_s_abk$3$(not in LMFDB)
3.3.ad_a_j$3$(not in LMFDB)
3.3.ad_j_as$3$(not in LMFDB)
3.3.a_a_a$3$(not in LMFDB)
3.3.a_j_a$3$(not in LMFDB)
3.3.d_a_aj$3$(not in LMFDB)
3.3.d_j_s$3$(not in LMFDB)
3.3.g_s_bk$3$(not in LMFDB)
3.3.j_bk_dd$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.ad_a_j$2$3.9.aj_cc_ahh
3.3.d_a_aj$2$3.9.aj_cc_ahh
3.3.j_bk_dd$2$3.9.aj_cc_ahh
3.3.ag_s_abk$3$(not in LMFDB)
3.3.ad_a_j$3$(not in LMFDB)
3.3.ad_j_as$3$(not in LMFDB)
3.3.a_a_a$3$(not in LMFDB)
3.3.a_j_a$3$(not in LMFDB)
3.3.d_a_aj$3$(not in LMFDB)
3.3.d_j_s$3$(not in LMFDB)
3.3.g_s_bk$3$(not in LMFDB)
3.3.j_bk_dd$3$(not in LMFDB)
3.3.ad_g_aj$4$(not in LMFDB)
3.3.d_g_j$4$(not in LMFDB)
3.3.a_a_aj$9$(not in LMFDB)
3.3.a_a_j$9$(not in LMFDB)
3.3.ad_ad_s$12$(not in LMFDB)
3.3.a_ad_a$12$(not in LMFDB)
3.3.a_g_a$12$(not in LMFDB)
3.3.d_ad_as$12$(not in LMFDB)
3.3.ad_d_a$24$(not in LMFDB)
3.3.a_d_a$24$(not in LMFDB)
3.3.d_d_a$24$(not in LMFDB)