Properties

Label 3.4.aj_bk_adk
Base field $\F_{2^{2}}$
Dimension $3$
$p$-rank $1$
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:  $\F_{2^{2}}$
Dimension:  $3$
L-polynomial:  $( 1 - 2 x )^{4}( 1 - x + 4 x^{2} )$
  $1 - 9 x + 36 x^{2} - 88 x^{3} + 144 x^{4} - 144 x^{5} + 64 x^{6}$
Frobenius angles:  $0$, $0$, $0$, $0$, $\pm0.419569376745$
Angle rank:  $1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $4$ $1944$ $182476$ $12150000$ $890274244$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-4$ $8$ $44$ $176$ $836$ $3848$ $16124$ $64736$ $259316$ $1042808$

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 $\F_{2^{2}}$.

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ae 2 $\times$ 1.4.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
3.4.ah_u_abo$2$3.16.aj_a_ge
3.4.ab_ae_i$2$3.16.aj_a_ge
3.4.b_ae_ai$2$3.16.aj_a_ge
3.4.h_u_bo$2$3.16.aj_a_ge
3.4.j_bk_dk$2$3.16.aj_a_ge
3.4.ad_g_aq$3$(not in LMFDB)
3.4.d_m_u$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.4.ah_u_abo$2$3.16.aj_a_ge
3.4.ab_ae_i$2$3.16.aj_a_ge
3.4.b_ae_ai$2$3.16.aj_a_ge
3.4.h_u_bo$2$3.16.aj_a_ge
3.4.j_bk_dk$2$3.16.aj_a_ge
3.4.ad_g_aq$3$(not in LMFDB)
3.4.d_m_u$3$(not in LMFDB)
3.4.af_q_abo$4$(not in LMFDB)
3.4.ad_i_ay$4$(not in LMFDB)
3.4.ab_m_ai$4$(not in LMFDB)
3.4.b_m_i$4$(not in LMFDB)
3.4.d_i_y$4$(not in LMFDB)
3.4.f_q_bo$4$(not in LMFDB)
3.4.b_g_m$5$(not in LMFDB)
3.4.ah_ba_acm$6$(not in LMFDB)
3.4.af_o_abg$6$(not in LMFDB)
3.4.af_u_abs$6$(not in LMFDB)
3.4.ad_m_au$6$(not in LMFDB)
3.4.ab_c_aq$6$(not in LMFDB)
3.4.ab_i_ae$6$(not in LMFDB)
3.4.b_c_q$6$(not in LMFDB)
3.4.b_i_e$6$(not in LMFDB)
3.4.d_g_q$6$(not in LMFDB)
3.4.f_o_bg$6$(not in LMFDB)
3.4.f_u_bs$6$(not in LMFDB)
3.4.h_ba_cm$6$(not in LMFDB)
3.4.ab_e_a$8$(not in LMFDB)
3.4.b_e_a$8$(not in LMFDB)
3.4.ad_k_au$10$(not in LMFDB)
3.4.ab_g_am$10$(not in LMFDB)
3.4.d_k_u$10$(not in LMFDB)
3.4.ad_o_ay$12$(not in LMFDB)
3.4.ab_a_e$12$(not in LMFDB)
3.4.ab_k_ai$12$(not in LMFDB)
3.4.b_a_ae$12$(not in LMFDB)
3.4.b_k_i$12$(not in LMFDB)
3.4.d_o_y$12$(not in LMFDB)