Properties

Label 4.2.af_n_ax_bi
Base field $\F_{2}$
Dimension $4$
$p$-rank $3$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

Downloads

Learn more

Invariants

Base field:  $\F_{2}$
Dimension:  $4$
L-polynomial:  $( 1 - 2 x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 7 x^{3} + 10 x^{4} - 12 x^{5} + 8 x^{6} )$
  $1 - 5 x + 13 x^{2} - 23 x^{3} + 34 x^{4} - 46 x^{5} + 52 x^{6} - 40 x^{7} + 16 x^{8}$
Frobenius angles:  $\pm0.105278500939$, $\pm0.250000000000$, $\pm0.316838792568$, $\pm0.641249159631$
Angle rank:  $3$ (numerical)
Jacobians:  $0$
Isomorphism classes:  1

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $2$ $460$ $3926$ $156400$ $1756522$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-2$ $6$ $10$ $30$ $48$ $42$ $110$ $286$ $532$ $1126$

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 $\times$ 3.2.ad_f_ah 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 $\times$ 3.16.f_br_fn. 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.ab_b_ad_g$2$4.4.b_h_ab_y
4.2.b_b_d_g$2$4.4.b_h_ab_y
4.2.f_n_x_bi$2$4.4.b_h_ab_y

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

TwistExtension degreeCommon base change
4.2.ab_b_ad_g$2$4.4.b_h_ab_y
4.2.b_b_d_g$2$4.4.b_h_ab_y
4.2.f_n_x_bi$2$4.4.b_h_ab_y
4.2.f_n_x_bi$4$(not in LMFDB)
4.2.ad_h_an_u$8$(not in LMFDB)
4.2.d_h_n_u$8$(not in LMFDB)