Properties

Label 4.3.ai_bh_adn_hb
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:  $( 1 - 3 x + 3 x^{2} )( 1 - 5 x + 15 x^{2} - 31 x^{3} + 45 x^{4} - 45 x^{5} + 27 x^{6} )$
Frobenius angles:  $\pm0.113296540390$, $\pm0.166666666667$, $\pm0.351823865540$, $\pm0.481790494592$
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 8281 732844 40585181 3533486887 315571420528 24860761811648 1917568997154629 153059630035416076 12278172330073214201

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -4 12 35 76 246 813 2369 6788 20069 59632

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 3.3.af_p_abf 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.bd_apb_abvut. 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.ac_d_ab_ad$2$(not in LMFDB)
4.3.c_d_b_ad$2$(not in LMFDB)
4.3.i_bh_dn_hb$2$(not in LMFDB)
4.3.af_s_abu_dm$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ac_d_ab_ad$2$(not in LMFDB)
4.3.c_d_b_ad$2$(not in LMFDB)
4.3.i_bh_dn_hb$2$(not in LMFDB)
4.3.af_s_abu_dm$3$(not in LMFDB)
4.3.f_s_bu_dm$6$(not in LMFDB)