Properties

Label 5.3.al_ci_aig_vj_abqd
Base field $\F_{3}$
Dimension $5$
$p$-rank $3$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{3}$
Dimension:  $5$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 5 x + 15 x^{2} - 31 x^{3} + 45 x^{4} - 45 x^{5} + 27 x^{6} )$
  $1 - 11 x + 60 x^{2} - 214 x^{3} + 555 x^{4} - 1095 x^{5} + 1665 x^{6} - 1926 x^{7} + 1620 x^{8} - 891 x^{9} + 243 x^{10}$
Frobenius angles:  $\pm0.113296540390$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.351823865540$, $\pm0.481790494592$
Angle rank:  $3$ (numerical)

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $7$ $57967$ $20519632$ $3693251471$ $957574946377$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-7$ $9$ $35$ $85$ $273$ $867$ $2450$ $6869$ $20069$ $59389$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{6}}$.

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 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 2 $\times$ 3.729.bd_apb_abvut. The endomorphism algebra for each factor is:
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
5.3.af_m_aq_j_d$2$(not in LMFDB)
5.3.ab_a_ac_d_p$2$(not in LMFDB)
5.3.b_a_c_d_ap$2$(not in LMFDB)
5.3.f_m_q_j_ad$2$(not in LMFDB)
5.3.l_ci_ig_vj_bqd$2$(not in LMFDB)
5.3.ai_bk_ael_kw_ava$3$(not in LMFDB)
5.3.af_m_aq_j_d$3$(not in LMFDB)
5.3.af_v_acj_fo_akq$3$(not in LMFDB)
5.3.ac_g_ah_g_ag$3$(not in LMFDB)
5.3.b_a_c_d_ap$3$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
5.3.af_m_aq_j_d$2$(not in LMFDB)
5.3.ab_a_ac_d_p$2$(not in LMFDB)
5.3.b_a_c_d_ap$2$(not in LMFDB)
5.3.f_m_q_j_ad$2$(not in LMFDB)
5.3.l_ci_ig_vj_bqd$2$(not in LMFDB)
5.3.ai_bk_ael_kw_ava$3$(not in LMFDB)
5.3.af_m_aq_j_d$3$(not in LMFDB)
5.3.af_v_acj_fo_akq$3$(not in LMFDB)
5.3.ac_g_ah_g_ag$3$(not in LMFDB)
5.3.b_a_c_d_ap$3$(not in LMFDB)
5.3.af_s_abu_dv_ahb$4$(not in LMFDB)
5.3.f_s_bu_dv_hb$4$(not in LMFDB)
5.3.c_g_h_g_g$6$(not in LMFDB)
5.3.f_v_cj_fo_kq$6$(not in LMFDB)
5.3.i_bk_el_kw_va$6$(not in LMFDB)
5.3.af_j_ab_abk_ds$12$(not in LMFDB)
5.3.f_j_b_abk_ads$12$(not in LMFDB)
5.3.af_p_abf_cc_adm$24$(not in LMFDB)
5.3.f_p_bf_cc_dm$24$(not in LMFDB)