Properties

Label 3.4.ak_bt_aem
Base field $\F_{2^{2}}$
Dimension $3$
$p$-rank $2$
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:  $( 1 - 2 x )^{2}( 1 - 3 x + 4 x^{2} )^{2}$
  $1 - 10 x + 45 x^{2} - 116 x^{3} + 180 x^{4} - 160 x^{5} + 64 x^{6}$
Frobenius angles:  $0$, $0$, $\pm0.230053456163$, $\pm0.230053456163$
Angle rank:  $1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $4$ $2304$ $268324$ $18662400$ $1125065764$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-5$ $7$ $67$ $287$ $1075$ $4063$ $15955$ $64127$ $259123$ $1044127$

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 $\times$ 1.4.ad 2 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.ae_d_e$2$3.16.ak_cn_amq
3.4.ac_ad_u$2$3.16.ak_cn_amq
3.4.c_ad_au$2$3.16.ak_cn_amq
3.4.e_d_ae$2$3.16.ak_cn_amq
3.4.k_bt_em$2$3.16.ak_cn_amq
3.4.ae_j_ao$3$(not in LMFDB)
3.4.ab_ad_e$3$(not in LMFDB)
3.4.f_p_bi$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.4.ae_d_e$2$3.16.ak_cn_amq
3.4.ac_ad_u$2$3.16.ak_cn_amq
3.4.c_ad_au$2$3.16.ak_cn_amq
3.4.e_d_ae$2$3.16.ak_cn_amq
3.4.k_bt_em$2$3.16.ak_cn_amq
3.4.ae_j_ao$3$(not in LMFDB)
3.4.ab_ad_e$3$(not in LMFDB)
3.4.f_p_bi$3$(not in LMFDB)
3.4.ag_v_abw$4$(not in LMFDB)
3.4.ae_f_ae$4$(not in LMFDB)
3.4.a_d_a$4$(not in LMFDB)
3.4.a_f_a$4$(not in LMFDB)
3.4.e_f_e$4$(not in LMFDB)
3.4.g_v_bw$4$(not in LMFDB)
3.4.ai_bh_ade$6$(not in LMFDB)
3.4.ah_v_abs$6$(not in LMFDB)
3.4.af_p_abi$6$(not in LMFDB)
3.4.ac_d_c$6$(not in LMFDB)
3.4.ab_d_ao$6$(not in LMFDB)
3.4.b_ad_ae$6$(not in LMFDB)
3.4.b_d_o$6$(not in LMFDB)
3.4.c_d_ac$6$(not in LMFDB)
3.4.e_j_o$6$(not in LMFDB)
3.4.h_v_bs$6$(not in LMFDB)
3.4.i_bh_de$6$(not in LMFDB)
3.4.ad_j_ay$12$(not in LMFDB)
3.4.ac_f_ac$12$(not in LMFDB)
3.4.c_f_c$12$(not in LMFDB)
3.4.d_j_y$12$(not in LMFDB)