Properties

Label 5.2.ag_r_abc_be_abg
Base field $\F_{2}$
Dimension $5$
$p$-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:  $\F_{2}$
Dimension:  $5$
L-polynomial:  $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 2 x + x^{2} + 2 x^{4} - 8 x^{5} + 8 x^{6} )$
  $1 - 6 x + 17 x^{2} - 28 x^{3} + 30 x^{4} - 32 x^{5} + 60 x^{6} - 112 x^{7} + 136 x^{8} - 96 x^{9} + 32 x^{10}$
Frobenius angles:  $\pm0.0693533547550$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.339907131295$, $\pm0.770553776540$
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:  $2$
Slopes:  $[0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $2$ $1100$ $65234$ $3740000$ $28143302$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-3$ $3$ $15$ $39$ $27$ $51$ $95$ $191$ $555$ $1043$

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 $\times$ 3.2.ac_b_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 $\times$ 3.16.g_r_ce. 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
5.2.ac_b_a_g_aq$2$(not in LMFDB)
5.2.ac_b_e_ac_a$2$(not in LMFDB)
5.2.c_b_ae_ac_a$2$(not in LMFDB)
5.2.c_b_a_g_q$2$(not in LMFDB)
5.2.g_r_bc_be_bg$2$(not in LMFDB)
5.2.a_ab_c_a_ai$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
5.2.ac_b_a_g_aq$2$(not in LMFDB)
5.2.ac_b_e_ac_a$2$(not in LMFDB)
5.2.c_b_ae_ac_a$2$(not in LMFDB)
5.2.c_b_a_g_q$2$(not in LMFDB)
5.2.g_r_bc_be_bg$2$(not in LMFDB)
5.2.a_ab_c_a_ai$3$(not in LMFDB)
5.2.ae_h_ak_q_ay$6$(not in LMFDB)
5.2.a_ab_ac_a_i$6$(not in LMFDB)
5.2.e_h_k_q_y$6$(not in LMFDB)
5.2.ae_j_ao_s_ay$8$(not in LMFDB)
5.2.ac_ad_i_c_aq$8$(not in LMFDB)
5.2.ac_f_ai_k_aq$8$(not in LMFDB)
5.2.a_b_ac_c_ai$8$(not in LMFDB)
5.2.a_b_c_c_i$8$(not in LMFDB)
5.2.c_ad_ai_c_q$8$(not in LMFDB)
5.2.c_f_i_k_q$8$(not in LMFDB)
5.2.e_j_o_s_y$8$(not in LMFDB)
5.2.a_ab_ac_a_i$12$(not in LMFDB)
5.2.ac_ab_e_e_aq$24$(not in LMFDB)
5.2.ac_d_ae_i_aq$24$(not in LMFDB)
5.2.c_ab_ae_e_q$24$(not in LMFDB)
5.2.c_d_e_i_q$24$(not in LMFDB)