Properties

Label 4.4.ak_bv_afm_mi
Base field $\F_{2^{2}}$
Dimension $4$
$p$-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:  $4$
L-polynomial:  $( 1 - 2 x )^{2}( 1 - 3 x + 4 x^{2} )( 1 - 3 x + 6 x^{2} - 12 x^{3} + 16 x^{4} )$
  $1 - 10 x + 47 x^{2} - 142 x^{3} + 320 x^{4} - 568 x^{5} + 752 x^{6} - 640 x^{7} + 256 x^{8}$
Frobenius angles:  $0$, $0$, $\pm0.150432950460$, $\pm0.230053456163$, $\pm0.544835058382$
Angle rank:  $3$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $16$ $43776$ $13256656$ $3979238400$ $1187129465776$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-5$ $11$ $49$ $239$ $1105$ $4223$ $16081$ $64767$ $261553$ $1045631$

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 $\times$ 2.4.ad_g 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
4.4.ae_f_ak_bg$2$(not in LMFDB)
4.4.ae_f_c_aq$2$(not in LMFDB)
4.4.ac_ab_k_aq$2$(not in LMFDB)
4.4.c_ab_ak_aq$2$(not in LMFDB)
4.4.e_f_ac_aq$2$(not in LMFDB)
4.4.e_f_k_bg$2$(not in LMFDB)
4.4.k_bv_fm_mi$2$(not in LMFDB)
4.4.ae_l_abc_cq$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
4.4.ae_f_ak_bg$2$(not in LMFDB)
4.4.ae_f_c_aq$2$(not in LMFDB)
4.4.ac_ab_k_aq$2$(not in LMFDB)
4.4.c_ab_ak_aq$2$(not in LMFDB)
4.4.e_f_ac_aq$2$(not in LMFDB)
4.4.e_f_k_bg$2$(not in LMFDB)
4.4.k_bv_fm_mi$2$(not in LMFDB)
4.4.ae_l_abc_cq$3$(not in LMFDB)
4.4.ag_x_aco_fw$4$(not in LMFDB)
4.4.a_f_ag_i$4$(not in LMFDB)
4.4.a_f_g_i$4$(not in LMFDB)
4.4.g_x_co_fw$4$(not in LMFDB)
4.4.ai_bj_aea_jc$6$(not in LMFDB)
4.4.ac_f_ai_u$6$(not in LMFDB)
4.4.ac_f_e_ae$6$(not in LMFDB)
4.4.c_f_ae_ae$6$(not in LMFDB)
4.4.c_f_i_u$6$(not in LMFDB)
4.4.e_l_bc_cq$6$(not in LMFDB)
4.4.i_bj_ea_jc$6$(not in LMFDB)