Properties

Label 3.4.ah_bb_acn
Base field $\F_{2^{2}}$
Dimension $3$
$p$-rank $3$
Ordinary yes
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:  $( 1 - x + 4 x^{2} )( 1 - 3 x + 4 x^{2} )^{2}$
  $1 - 7 x + 27 x^{2} - 65 x^{3} + 108 x^{4} - 112 x^{5} + 64 x^{6}$
Frobenius angles:  $\pm0.230053456163$, $\pm0.230053456163$, $\pm0.419569376745$
Angle rank:  $2$ (numerical)

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $16$ $6144$ $416176$ $19906560$ $1128577936$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-2$ $22$ $94$ $302$ $1078$ $4198$ $16462$ $64862$ $259366$ $1044502$

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.ad 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.af_p_abf$2$3.16.f_bj_gl
3.4.ab_d_b$2$3.16.f_bj_gl
3.4.b_d_ab$2$3.16.f_bj_gl
3.4.f_p_bf$2$3.16.f_bj_gl
3.4.h_bb_cn$2$3.16.f_bj_gl
3.4.c_g_t$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.4.af_p_abf$2$3.16.f_bj_gl
3.4.ab_d_b$2$3.16.f_bj_gl
3.4.b_d_ab$2$3.16.f_bj_gl
3.4.f_p_bf$2$3.16.f_bj_gl
3.4.h_bb_cn$2$3.16.f_bj_gl
3.4.c_g_t$3$(not in LMFDB)
3.4.ab_f_ab$4$(not in LMFDB)
3.4.b_f_b$4$(not in LMFDB)
3.4.ae_m_abd$6$(not in LMFDB)
3.4.ac_g_at$6$(not in LMFDB)
3.4.e_m_bd$6$(not in LMFDB)