Properties

Label 3.4.ag_y_ace
Base field $\F_{2^{2}}$
Dimension $3$
$p$-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:  $\F_{2^{2}}$
Dimension:  $3$
L-polynomial:  $( 1 - 2 x + 4 x^{2} )^{3}$
  $1 - 6 x + 24 x^{2} - 56 x^{3} + 96 x^{4} - 96 x^{5} + 64 x^{6}$
Frobenius angles:  $\pm0.333333333333$, $\pm0.333333333333$, $\pm0.333333333333$
Angle rank:  $0$ (numerical)

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:  $[1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $27$ $9261$ $531441$ $20346417$ $979146657$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $29$ $113$ $305$ $929$ $3713$ $16001$ $66305$ $265217$ $1051649$

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 $\F_{2^{6}}$.

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ac 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
Endomorphism algebra over $\overline{\F}_{2^{2}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.q 3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
3.4.ac_i_ai$2$3.16.m_ds_rg
3.4.c_i_i$2$3.16.m_ds_rg
3.4.g_y_ce$2$3.16.m_ds_rg
3.4.a_a_q$3$(not in LMFDB)
3.4.g_m_q$3$(not in LMFDB)
3.4.m_ci_ge$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.4.ac_i_ai$2$3.16.m_ds_rg
3.4.c_i_i$2$3.16.m_ds_rg
3.4.g_y_ce$2$3.16.m_ds_rg
3.4.a_a_q$3$(not in LMFDB)
3.4.g_m_q$3$(not in LMFDB)
3.4.m_ci_ge$3$(not in LMFDB)
3.4.ac_a_i$4$(not in LMFDB)
3.4.c_a_ai$4$(not in LMFDB)
3.4.am_ci_age$6$(not in LMFDB)
3.4.ak_bs_aei$6$(not in LMFDB)
3.4.ai_bg_adc$6$(not in LMFDB)
3.4.ag_m_aq$6$(not in LMFDB)
3.4.ae_ae_bg$6$(not in LMFDB)
3.4.ae_i_aq$6$(not in LMFDB)
3.4.ac_ae_q$6$(not in LMFDB)
3.4.a_a_aq$6$(not in LMFDB)
3.4.c_ae_aq$6$(not in LMFDB)
3.4.e_ae_abg$6$(not in LMFDB)
3.4.e_i_q$6$(not in LMFDB)
3.4.i_bg_dc$6$(not in LMFDB)
3.4.k_bs_ei$6$(not in LMFDB)
3.4.a_a_ai$9$(not in LMFDB)
3.4.ai_bc_acm$12$(not in LMFDB)
3.4.ag_u_abw$12$(not in LMFDB)
3.4.ae_a_q$12$(not in LMFDB)
3.4.ae_m_abg$12$(not in LMFDB)
3.4.ae_q_abg$12$(not in LMFDB)
3.4.ac_e_aq$12$(not in LMFDB)
3.4.ac_m_aq$12$(not in LMFDB)
3.4.a_ae_a$12$(not in LMFDB)
3.4.a_a_a$12$(not in LMFDB)
3.4.a_i_a$12$(not in LMFDB)
3.4.a_m_a$12$(not in LMFDB)
3.4.c_e_q$12$(not in LMFDB)
3.4.c_m_q$12$(not in LMFDB)
3.4.e_a_aq$12$(not in LMFDB)
3.4.e_m_bg$12$(not in LMFDB)
3.4.e_q_bg$12$(not in LMFDB)
3.4.g_u_bw$12$(not in LMFDB)
3.4.i_bc_cm$12$(not in LMFDB)
3.4.ae_m_ay$15$(not in LMFDB)
3.4.c_a_a$15$(not in LMFDB)
3.4.a_a_i$18$(not in LMFDB)
3.4.ae_e_a$24$(not in LMFDB)
3.4.ac_e_a$24$(not in LMFDB)
3.4.a_e_a$24$(not in LMFDB)
3.4.c_e_a$24$(not in LMFDB)
3.4.e_e_a$24$(not in LMFDB)
3.4.ag_q_abg$30$(not in LMFDB)
3.4.ac_a_a$30$(not in LMFDB)
3.4.a_e_ai$30$(not in LMFDB)
3.4.a_e_i$30$(not in LMFDB)
3.4.e_m_y$30$(not in LMFDB)
3.4.g_q_bg$30$(not in LMFDB)
3.4.ac_i_aq$60$(not in LMFDB)
3.4.c_i_q$60$(not in LMFDB)