Properties

Label 3.4.ah_w_abw
Base field $\F_{2^{2}}$
Dimension $3$
$p$-rank $1$
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:  $( 1 - 2 x )^{2}( 1 - 3 x + 6 x^{2} - 12 x^{3} + 16 x^{4} )$
  $1 - 7 x + 22 x^{2} - 48 x^{3} + 88 x^{4} - 112 x^{5} + 64 x^{6}$
Frobenius angles:  $0$, $0$, $\pm0.150432950460$, $\pm0.544835058382$
Angle rank:  $2$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $8$ $2736$ $179144$ $13816800$ $1097162168$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-2$ $12$ $40$ $208$ $1048$ $4176$ $16168$ $65216$ $262552$ $1046832$

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 $\times$ 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
3.4.ab_ac_a$2$3.16.af_am_ey
3.4.b_ac_a$2$3.16.af_am_ey
3.4.h_w_bw$2$3.16.af_am_ey
3.4.ab_e_am$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.4.ab_ac_a$2$3.16.af_am_ey
3.4.b_ac_a$2$3.16.af_am_ey
3.4.h_w_bw$2$3.16.af_am_ey
3.4.ab_e_am$3$(not in LMFDB)
3.4.ad_k_ay$4$(not in LMFDB)
3.4.d_k_y$4$(not in LMFDB)
3.4.af_q_abk$6$(not in LMFDB)
3.4.b_e_m$6$(not in LMFDB)
3.4.f_q_bk$6$(not in LMFDB)