Properties

Label 4.2.ae_h_af_c
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

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Invariants

Base field:  $\F_{2}$
Dimension:  $4$
L-polynomial:  $( 1 - 2 x + 2 x^{2} )( 1 - 2 x + x^{2} + x^{3} + 2 x^{4} - 8 x^{5} + 8 x^{6} )$
  $1 - 4 x + 7 x^{2} - 5 x^{3} + 2 x^{4} - 10 x^{5} + 28 x^{6} - 32 x^{7} + 16 x^{8}$
Frobenius angles:  $\pm0.132091856901$, $\pm0.250000000000$, $\pm0.309487084859$, $\pm0.780459932197$
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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $3$ $315$ $8541$ $247275$ $1097283$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $3$ $14$ $39$ $34$ $72$ $118$ $231$ $599$ $1068$

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.ac_b_b 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.o_dx_sl. 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.a_ab_ab_g$2$4.4.ac_n_av_cu
4.2.a_ab_b_g$2$4.4.ac_n_av_cu
4.2.e_h_f_c$2$4.4.ac_n_av_cu

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

TwistExtension degreeCommon base change
4.2.a_ab_ab_g$2$4.4.ac_n_av_cu
4.2.a_ab_b_g$2$4.4.ac_n_av_cu
4.2.e_h_f_c$2$4.4.ac_n_av_cu
4.2.ac_d_ad_e$8$(not in LMFDB)
4.2.c_d_d_e$8$(not in LMFDB)