Properties

Label 4.4.ak_bw_afv_nk
Base field $\F_{2^{2}}$
Dimension $4$
$p$-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:  $\F_{2^{2}}$
Dimension:  $4$
L-polynomial:  $( 1 - 2 x )^{2}( 1 - 6 x + 20 x^{2} - 47 x^{3} + 80 x^{4} - 96 x^{5} + 64 x^{6} )$
  $1 - 10 x + 48 x^{2} - 151 x^{3} + 348 x^{4} - 604 x^{5} + 768 x^{6} - 640 x^{7} + 256 x^{8}$
Frobenius angles:  $0$, $0$, $\pm0.0842195913666$, $\pm0.309444685979$, $\pm0.509145309743$
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:  $[0, 0, 0, 1/2, 1/2, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $16$ $45216$ $13513024$ $3429633600$ $1018658016496$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-5$ $13$ $52$ $201$ $945$ $3982$ $15927$ $64753$ $262924$ $1052253$

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 $\times$ 3.4.ag_u_abv 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.ac_a_j_abc$2$(not in LMFDB)
4.4.c_a_aj_abc$2$(not in LMFDB)
4.4.k_bw_fv_nk$2$(not in LMFDB)
4.4.ae_m_abf_co$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
4.4.ac_a_j_abc$2$(not in LMFDB)
4.4.c_a_aj_abc$2$(not in LMFDB)
4.4.k_bw_fv_nk$2$(not in LMFDB)
4.4.ae_m_abf_co$3$(not in LMFDB)
4.4.ag_y_act_ge$4$(not in LMFDB)
4.4.g_y_ct_ge$4$(not in LMFDB)
4.4.ai_bk_aeh_ju$6$(not in LMFDB)
4.4.e_m_bf_co$6$(not in LMFDB)
4.4.i_bk_eh_ju$6$(not in LMFDB)