Properties

Label 4.2.ae_e_i_ay
Base field $\F_{2}$
Dimension $4$
$p$-rank $0$
Ordinary no
Supersingular yes
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:  $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 2 x^{2} )^{2}$
  $1 - 4 x + 4 x^{2} + 8 x^{3} - 24 x^{4} + 16 x^{5} + 16 x^{6} - 32 x^{7} + 16 x^{8}$
Frobenius angles:  $0$, $0$, $\pm0.250000000000$, $\pm0.250000000000$, $1$, $1$
Angle rank:  $0$ (numerical)
Jacobians:  $0$

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1$ $25$ $8281$ $50625$ $1615441$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $-3$ $17$ $17$ $49$ $33$ $97$ $129$ $449$ $897$

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 $\F_{2^{8}}$.

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.ac 2 $\times$ 2.2.a_ae 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 $\F_{2^{8}}$ is 1.256.abg 4 and its endomorphism algebra is $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
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_ae_a_i$2$4.4.ai_bg_ads_iq
4.2.e_e_ai_ay$2$4.4.ai_bg_ads_iq
4.2.ae_k_aq_y$3$(not in LMFDB)
4.2.c_ac_ae_a$3$(not in LMFDB)
4.2.c_e_i_m$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
4.2.a_ae_a_i$2$4.4.ai_bg_ads_iq
4.2.e_e_ai_ay$2$4.4.ai_bg_ads_iq
4.2.ae_k_aq_y$3$(not in LMFDB)
4.2.c_ac_ae_a$3$(not in LMFDB)
4.2.c_e_i_m$3$(not in LMFDB)
4.2.ae_m_ay_bo$4$(not in LMFDB)
4.2.a_e_a_i$4$(not in LMFDB)
4.2.e_m_y_bo$4$(not in LMFDB)
4.2.ac_ac_e_a$6$(not in LMFDB)
4.2.ac_e_ai_m$6$(not in LMFDB)
4.2.a_c_a_i$6$(not in LMFDB)
4.2.e_k_q_y$6$(not in LMFDB)
4.2.ai_bg_adc_fg$8$(not in LMFDB)
4.2.ag_u_abs_cu$8$(not in LMFDB)
4.2.ae_i_ai_i$8$(not in LMFDB)
4.2.ac_a_e_ai$8$(not in LMFDB)
4.2.ac_e_ae_i$8$(not in LMFDB)
4.2.ac_i_am_y$8$(not in LMFDB)
4.2.a_ai_a_y$8$(not in LMFDB)
4.2.a_a_a_ai$8$(not in LMFDB)
4.2.a_a_a_i$8$(not in LMFDB)
4.2.a_i_a_y$8$(not in LMFDB)
4.2.c_a_ae_ai$8$(not in LMFDB)
4.2.c_e_e_i$8$(not in LMFDB)
4.2.c_i_m_y$8$(not in LMFDB)
4.2.e_i_i_i$8$(not in LMFDB)
4.2.g_u_bs_cu$8$(not in LMFDB)
4.2.i_bg_dc_fg$8$(not in LMFDB)
4.2.ae_g_a_ai$12$(not in LMFDB)
4.2.ac_a_a_e$12$(not in LMFDB)
4.2.ac_g_am_q$12$(not in LMFDB)
4.2.a_ac_a_i$12$(not in LMFDB)
4.2.c_a_a_e$12$(not in LMFDB)
4.2.c_g_m_q$12$(not in LMFDB)
4.2.e_g_a_ai$12$(not in LMFDB)
4.2.a_a_a_a$16$(not in LMFDB)
4.2.ag_s_abk_ce$24$(not in LMFDB)
4.2.ae_i_aq_bc$24$(not in LMFDB)
4.2.ae_k_au_bg$24$(not in LMFDB)
4.2.ac_c_ae_i$24$(not in LMFDB)
4.2.ac_c_a_a$24$(not in LMFDB)
4.2.ac_c_e_ai$24$(not in LMFDB)
4.2.ac_g_ai_q$24$(not in LMFDB)
4.2.a_ag_a_q$24$(not in LMFDB)
4.2.a_ae_a_m$24$(not in LMFDB)
4.2.a_ac_a_a$24$(not in LMFDB)
4.2.a_a_a_ae$24$(not in LMFDB)
4.2.a_a_a_e$24$(not in LMFDB)
4.2.a_c_ae_a$24$(not in LMFDB)
4.2.a_c_a_a$24$(not in LMFDB)
4.2.a_c_e_a$24$(not in LMFDB)
4.2.a_e_a_m$24$(not in LMFDB)
4.2.a_g_a_q$24$(not in LMFDB)
4.2.c_c_ae_ai$24$(not in LMFDB)
4.2.c_c_a_a$24$(not in LMFDB)
4.2.c_c_e_i$24$(not in LMFDB)
4.2.c_g_i_q$24$(not in LMFDB)
4.2.e_i_q_bc$24$(not in LMFDB)
4.2.e_k_u_bg$24$(not in LMFDB)
4.2.g_s_bk_ce$24$(not in LMFDB)
4.2.ac_c_a_ae$40$(not in LMFDB)
4.2.a_ac_a_e$40$(not in LMFDB)
4.2.a_c_a_e$40$(not in LMFDB)
4.2.c_c_a_ae$40$(not in LMFDB)