Properties

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

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Invariants

Base field:  $\F_{3}$
Dimension:  $4$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 - 4 x + 9 x^{2} - 17 x^{3} + 27 x^{4} - 36 x^{5} + 27 x^{6} )$
Frobenius angles:  $\pm0.0653366913680$, $\pm0.166666666667$, $\pm0.328985474983$, $\pm0.609104440316$
Angle rank:  $3$ (numerical)

This isogeny class is not simple.

Newton polygon

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1/2, 1/2, 1, 1, 1]$

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
$A(\F_{q^r})$ 7 5929 400624 45706661 3908457392 266033565952 22379220236513 1942216848112541 152965130185763728 12119527096636999424

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 9 21 85 272 687 2139 6869 20055 58864

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 3.3.ae_j_ar 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 $\times$ 3.729.adt_hiy_ajlhg. The endomorphism algebra for each factor is:
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.ab_a_ac_d$2$(not in LMFDB)
4.3.b_a_c_d$2$(not in LMFDB)
4.3.h_y_ce_eb$2$(not in LMFDB)
4.3.ae_m_abd_cc$3$(not in LMFDB)
4.3.ab_a_ac_d$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ab_a_ac_d$2$(not in LMFDB)
4.3.b_a_c_d$2$(not in LMFDB)
4.3.h_y_ce_eb$2$(not in LMFDB)
4.3.ae_m_abd_cc$3$(not in LMFDB)
4.3.ab_a_ac_d$3$(not in LMFDB)
4.3.e_m_bd_cc$6$(not in LMFDB)