Properties

Label 4.2.af_m_as_y
Base field $\F_{2}$
Dimension $4$
$p$-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:  $\F_{2}$
Dimension:  $4$
L-polynomial:  $( 1 - 2 x + 2 x^{2} )^{2}( 1 - x - 2 x^{3} + 4 x^{4} )$
  $1 - 5 x + 12 x^{2} - 18 x^{3} + 24 x^{4} - 36 x^{5} + 48 x^{6} - 40 x^{7} + 16 x^{8}$
Frobenius angles:  $\pm0.139386741866$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.686170398078$
Angle rank:  $2$ (numerical)
Jacobians:  $0$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $2$ $400$ $4394$ $260000$ $2356762$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-2$ $4$ $10$ $40$ $58$ $64$ $138$ $224$ $442$ $1104$

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 $\F_{2^{4}}$.

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.ac 2 $\times$ 2.2.ab_a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 2 $\times$ 2.16.h_bo. The endomorphism algebra for each factor is:
Remainder of endomorphism lattice by field

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
4.2.ad_e_c_ai$2$4.4.ab_m_am_cm
4.2.ab_a_ac_i$2$4.4.ab_m_am_cm
4.2.b_a_c_i$2$4.4.ab_m_am_cm
4.2.d_e_ac_ai$2$4.4.ab_m_am_cm
4.2.f_m_s_y$2$4.4.ab_m_am_cm
4.2.b_a_a_a$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
4.2.ad_e_c_ai$2$4.4.ab_m_am_cm
4.2.ab_a_ac_i$2$4.4.ab_m_am_cm
4.2.b_a_c_i$2$4.4.ab_m_am_cm
4.2.d_e_ac_ai$2$4.4.ab_m_am_cm
4.2.f_m_s_y$2$4.4.ab_m_am_cm
4.2.b_a_a_a$3$(not in LMFDB)
4.2.ad_e_ai_q$6$(not in LMFDB)
4.2.ab_a_a_a$6$(not in LMFDB)
4.2.d_e_i_q$6$(not in LMFDB)
4.2.ad_g_ak_q$8$(not in LMFDB)
4.2.ab_ae_c_i$8$(not in LMFDB)
4.2.ab_c_c_a$8$(not in LMFDB)
4.2.ab_e_ag_i$8$(not in LMFDB)
4.2.b_ae_ac_i$8$(not in LMFDB)
4.2.b_c_ac_a$8$(not in LMFDB)
4.2.b_e_g_i$8$(not in LMFDB)
4.2.d_g_k_q$8$(not in LMFDB)
4.2.ad_e_ai_q$12$(not in LMFDB)
4.2.ab_ac_a_i$24$(not in LMFDB)
4.2.ab_c_ae_i$24$(not in LMFDB)
4.2.b_ac_a_i$24$(not in LMFDB)
4.2.b_c_e_i$24$(not in LMFDB)