Properties

Label 3.5.ak_bw_afg
Base field $\F_{5}$
Dimension $3$
$p$-rank $3$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{5}$
Dimension:  $3$
L-polynomial:  $( 1 - 4 x + 5 x^{2} )( 1 - 3 x + 5 x^{2} )^{2}$
  $1 - 10 x + 48 x^{2} - 136 x^{3} + 240 x^{4} - 250 x^{5} + 125 x^{6}$
Frobenius angles:  $\pm0.147583617650$, $\pm0.265942140215$, $\pm0.265942140215$
Angle rank:  $2$ (numerical)
Jacobians:  $0$

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $18$ $14580$ $2529792$ $291600000$ $32441027778$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-4$ $22$ $158$ $738$ $3316$ $15712$ $77668$ $389378$ $1952726$ $9772102$

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 $\F_{5}$.

Endomorphism algebra over $\F_{5}$
The isogeny class factors as 1.5.ae $\times$ 1.5.ad 2 and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
3.5.ae_g_ae$2$3.25.ae_cm_ahy
3.5.ac_a_q$2$3.25.ae_cm_ahy
3.5.c_a_aq$2$3.25.ae_cm_ahy
3.5.e_g_e$2$3.25.ae_cm_ahy
3.5.k_bw_fg$2$3.25.ae_cm_ahy
3.5.ab_ad_o$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.5.ae_g_ae$2$3.25.ae_cm_ahy
3.5.ac_a_q$2$3.25.ae_cm_ahy
3.5.c_a_aq$2$3.25.ae_cm_ahy
3.5.e_g_e$2$3.25.ae_cm_ahy
3.5.k_bw_fg$2$3.25.ae_cm_ahy
3.5.ab_ad_o$3$(not in LMFDB)
3.5.ai_bk_adu$4$(not in LMFDB)
3.5.ae_e_e$4$(not in LMFDB)
3.5.ae_m_aw$4$(not in LMFDB)
3.5.ac_e_c$4$(not in LMFDB)
3.5.ac_g_ac$4$(not in LMFDB)
3.5.c_e_ac$4$(not in LMFDB)
3.5.c_g_c$4$(not in LMFDB)
3.5.e_e_ae$4$(not in LMFDB)
3.5.e_m_w$4$(not in LMFDB)
3.5.i_bk_du$4$(not in LMFDB)
3.5.ah_v_abu$6$(not in LMFDB)
3.5.b_ad_ao$6$(not in LMFDB)
3.5.h_v_bu$6$(not in LMFDB)
3.5.af_p_abm$12$(not in LMFDB)
3.5.ab_d_aw$12$(not in LMFDB)
3.5.b_d_w$12$(not in LMFDB)
3.5.f_p_bm$12$(not in LMFDB)