Properties

Label 4.3.aj_bn_aee_ii
Base Field $\F_{3}$
Dimension $4$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

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

This isogeny class is not simple.

Newton polygon

This isogeny class is supersingular.

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

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
$A(\F_{q^r})$ 4 5488 614656 48228544 4856212684 377801998336 25559408866492 1876172350124800 150125140011540736 12107841492722430448

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -5 7 28 91 325 946 2431 6643 19684 58807

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 3 $\times$ 1.3.a 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 4 and its endomorphism algebra is $\mathrm{M}_{4}(B)$, 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
4.3.ad_d_a_a$2$(not in LMFDB)
4.3.d_d_a_a$2$(not in LMFDB)
4.3.j_bn_ee_ii$2$(not in LMFDB)
4.3.am_co_aii_rr$3$(not in LMFDB)
4.3.ag_m_a_abb$3$(not in LMFDB)
4.3.ag_v_acc_ee$3$(not in LMFDB)
4.3.ad_d_a_a$3$(not in LMFDB)
4.3.ad_m_abb_cc$3$(not in LMFDB)
4.3.a_ag_a_bb$3$(not in LMFDB)
4.3.a_d_a_a$3$(not in LMFDB)
4.3.a_m_a_cc$3$(not in LMFDB)
4.3.d_d_a_a$3$(not in LMFDB)
4.3.d_m_bb_cc$3$(not in LMFDB)
4.3.g_m_a_abb$3$(not in LMFDB)
4.3.g_v_cc_ee$3$(not in LMFDB)
4.3.j_bn_ee_ii$3$(not in LMFDB)
4.3.m_co_ii_rr$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ad_d_a_a$2$(not in LMFDB)
4.3.d_d_a_a$2$(not in LMFDB)
4.3.j_bn_ee_ii$2$(not in LMFDB)
4.3.am_co_aii_rr$3$(not in LMFDB)
4.3.ag_m_a_abb$3$(not in LMFDB)
4.3.ag_v_acc_ee$3$(not in LMFDB)
4.3.ad_d_a_a$3$(not in LMFDB)
4.3.ad_m_abb_cc$3$(not in LMFDB)
4.3.a_ag_a_bb$3$(not in LMFDB)
4.3.a_d_a_a$3$(not in LMFDB)
4.3.a_m_a_cc$3$(not in LMFDB)
4.3.d_d_a_a$3$(not in LMFDB)
4.3.d_m_bb_cc$3$(not in LMFDB)
4.3.g_m_a_abb$3$(not in LMFDB)
4.3.g_v_cc_ee$3$(not in LMFDB)
4.3.j_bn_ee_ii$3$(not in LMFDB)
4.3.m_co_ii_rr$3$(not in LMFDB)
4.3.ad_j_as_bk$4$(not in LMFDB)
4.3.d_j_s_bk$4$(not in LMFDB)
4.3.j_bn_ee_ii$6$(not in LMFDB)
4.3.ad_d_aj_bb$9$(not in LMFDB)
4.3.ad_d_j_abb$9$(not in LMFDB)
4.3.a_d_aj_a$9$(not in LMFDB)
4.3.a_d_j_a$9$(not in LMFDB)
4.3.d_d_aj_abb$9$(not in LMFDB)
4.3.d_d_j_bb$9$(not in LMFDB)
4.3.ag_j_s_acu$12$(not in LMFDB)
4.3.ag_s_abk_cl$12$(not in LMFDB)
4.3.ad_a_j_as$12$(not in LMFDB)
4.3.a_am_a_cc$12$(not in LMFDB)
4.3.a_aj_a_bk$12$(not in LMFDB)
4.3.a_ad_a_a$12$(not in LMFDB)
4.3.a_a_a_as$12$(not in LMFDB)
4.3.a_a_a_j$12$(not in LMFDB)
4.3.a_g_a_bb$12$(not in LMFDB)
4.3.a_j_a_bk$12$(not in LMFDB)
4.3.d_a_aj_as$12$(not in LMFDB)
4.3.g_j_as_acu$12$(not in LMFDB)
4.3.g_s_bk_cl$12$(not in LMFDB)
4.3.ad_g_aj_j$15$(not in LMFDB)
4.3.a_ad_a_j$15$(not in LMFDB)
4.3.d_g_j_j$15$(not in LMFDB)
4.3.ag_p_as_s$24$(not in LMFDB)
4.3.ad_g_aj_s$24$(not in LMFDB)
4.3.a_ag_a_s$24$(not in LMFDB)
4.3.a_ad_a_s$24$(not in LMFDB)
4.3.a_a_a_aj$24$(not in LMFDB)
4.3.a_a_a_s$24$(not in LMFDB)
4.3.a_d_a_s$24$(not in LMFDB)
4.3.a_g_a_s$24$(not in LMFDB)
4.3.d_g_j_s$24$(not in LMFDB)
4.3.g_p_s_s$24$(not in LMFDB)
4.3.a_a_a_a$48$(not in LMFDB)
4.3.a_d_a_j$60$(not in LMFDB)