Properties

Label 3.2.af_o_ay
Base field $\F_{2}$
Dimension $3$
$p$-rank $1$
Ordinary No
Supersingular No
Simple No
Geometrically simple No
Primitive Yes
Principally polarizable Yes
Contains a Jacobian No

Learn more about

Invariants

Base field:  $\F_{2}$
Dimension:  $3$
L-polynomial:  $( 1 - x + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{2}$
Frobenius angles:  $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.384973271919$
Angle rank:  $1$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2 200 2366 10000 36982 236600 1813198 14580000 119844998 1017005000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 8 22 32 38 56 110 224 454 968

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.ac 2 $\times$ 1.2.ab 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.ab $\times$ 1.16.i 2 . The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{2^{4}}$.
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
3.2.ad_g_ai$2$3.4.d_m_y
3.2.ab_c_a$2$3.4.d_m_y
3.2.b_c_a$2$3.4.d_m_y
3.2.d_g_i$2$3.4.d_m_y
3.2.f_o_y$2$3.4.d_m_y
3.2.b_c_g$3$3.8.n_dc_lc
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.2.ad_g_ai$2$3.4.d_m_y
3.2.ab_c_a$2$3.4.d_m_y
3.2.b_c_a$2$3.4.d_m_y
3.2.d_g_i$2$3.4.d_m_y
3.2.f_o_y$2$3.4.d_m_y
3.2.b_c_g$3$3.8.n_dc_lc
3.2.ad_g_ai$4$3.16.p_ds_qa
3.2.ab_c_a$4$3.16.p_ds_qa
3.2.b_c_a$4$3.16.p_ds_qa
3.2.d_g_i$4$3.16.p_ds_qa
3.2.f_o_y$4$3.16.p_ds_qa
3.2.ad_g_ak$6$(not in LMFDB)
3.2.ab_c_ag$6$(not in LMFDB)
3.2.d_g_k$6$(not in LMFDB)
3.2.ad_i_am$8$(not in LMFDB)
3.2.ab_ac_e$8$(not in LMFDB)
3.2.ab_e_ae$8$(not in LMFDB)
3.2.ab_g_ae$8$(not in LMFDB)
3.2.b_ac_ae$8$(not in LMFDB)
3.2.b_e_e$8$(not in LMFDB)
3.2.b_g_e$8$(not in LMFDB)
3.2.d_i_m$8$(not in LMFDB)
3.2.ad_g_ak$12$(not in LMFDB)
3.2.ab_c_ag$12$(not in LMFDB)
3.2.b_c_g$12$(not in LMFDB)
3.2.d_g_k$12$(not in LMFDB)
3.2.ab_a_c$24$(not in LMFDB)
3.2.ab_e_ac$24$(not in LMFDB)
3.2.b_a_ac$24$(not in LMFDB)
3.2.b_e_c$24$(not in LMFDB)