Properties

Label 3.4.ag_v_aby
Base field $\F_{2^{2}}$
Dimension $3$
$p$-rank $2$
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:  $( 1 - 2 x + 4 x^{2} )( 1 - 4 x + 9 x^{2} - 16 x^{3} + 16 x^{4} )$
  $1 - 6 x + 21 x^{2} - 50 x^{3} + 84 x^{4} - 96 x^{5} + 64 x^{6}$
Frobenius angles:  $\pm0.117169895439$, $\pm0.333333333333$, $\pm0.478661301576$
Angle rank:  $2$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $18$ $5796$ $319302$ $15672384$ $1038842838$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $23$ $77$ $239$ $989$ $4163$ $16589$ $65855$ $263597$ $1052963$

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 $\F_{2^{6}}$.

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ac $\times$ 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:
Endomorphism algebra over $\overline{\F}_{2^{2}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.q $\times$ 2.64.ae_eb. 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.ac_f_ao$2$3.16.g_j_e
3.4.c_f_o$2$3.16.g_j_e
3.4.g_v_by$2$3.16.g_j_e
3.4.a_ad_e$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.4.ac_f_ao$2$3.16.g_j_e
3.4.c_f_o$2$3.16.g_j_e
3.4.g_v_by$2$3.16.g_j_e
3.4.a_ad_e$3$(not in LMFDB)
3.4.ai_bd_acq$6$(not in LMFDB)
3.4.a_ad_ae$6$(not in LMFDB)
3.4.i_bd_cq$6$(not in LMFDB)
3.4.ae_n_abg$12$(not in LMFDB)
3.4.e_n_bg$12$(not in LMFDB)