Properties

Label 4.4.al_cf_ahf_qu
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:  $( 1 - 2 x )^{2}( 1 - 3 x + 4 x^{2} )( 1 - 4 x + 9 x^{2} - 16 x^{3} + 16 x^{4} )$
  $1 - 11 x + 57 x^{2} - 187 x^{3} + 436 x^{4} - 748 x^{5} + 912 x^{6} - 704 x^{7} + 256 x^{8}$
Frobenius angles:  $0$, $0$, $\pm0.117169895439$, $\pm0.230053456163$, $\pm0.478661301576$
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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $12$ $39744$ $14293692$ $3720038400$ $1087805499132$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-6$ $10$ $54$ $222$ $1014$ $4210$ $16374$ $64638$ $260550$ $1048690$

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.ae_j 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.af_j_an_bc$2$(not in LMFDB)
4.4.ad_b_f_am$2$(not in LMFDB)
4.4.ad_b_n_abk$2$(not in LMFDB)
4.4.d_b_an_abk$2$(not in LMFDB)
4.4.d_b_af_am$2$(not in LMFDB)
4.4.f_j_n_bc$2$(not in LMFDB)
4.4.l_cf_hf_qu$2$(not in LMFDB)
4.4.af_p_abl_de$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
4.4.af_j_an_bc$2$(not in LMFDB)
4.4.ad_b_f_am$2$(not in LMFDB)
4.4.ad_b_n_abk$2$(not in LMFDB)
4.4.d_b_an_abk$2$(not in LMFDB)
4.4.d_b_af_am$2$(not in LMFDB)
4.4.f_j_n_bc$2$(not in LMFDB)
4.4.l_cf_hf_qu$2$(not in LMFDB)
4.4.af_p_abl_de$3$(not in LMFDB)
4.4.ah_bd_adj_hs$4$(not in LMFDB)
4.4.ab_f_aj_i$4$(not in LMFDB)
4.4.b_f_j_i$4$(not in LMFDB)
4.4.h_bd_dj_hs$4$(not in LMFDB)
4.4.aj_br_afh_mg$6$(not in LMFDB)
4.4.ad_h_al_s$6$(not in LMFDB)
4.4.ab_d_h_ac$6$(not in LMFDB)
4.4.b_d_ah_ac$6$(not in LMFDB)
4.4.d_h_l_s$6$(not in LMFDB)
4.4.f_p_bl_de$6$(not in LMFDB)
4.4.j_br_fh_mg$6$(not in LMFDB)