Properties

Label 3.5.ai_bg_adg
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:  $( 1 - 4 x + 5 x^{2} )( 1 - 4 x + 11 x^{2} - 20 x^{3} + 25 x^{4} )$
  $1 - 8 x + 32 x^{2} - 84 x^{3} + 160 x^{4} - 200 x^{5} + 125 x^{6}$
Frobenius angles:  $\pm0.147583617650$, $\pm0.185749715683$, $\pm0.480916950984$
Angle rank:  $1$ (numerical)
Jacobians:  $0$
Isomorphism classes:  2

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $26$ $15860$ $2061800$ $244117120$ $32038741306$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-2$ $26$ $130$ $626$ $3278$ $16328$ $79238$ $390626$ $1951690$ $9765626$

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 $\F_{5^{6}}$.

Endomorphism algebra over $\F_{5}$
The isogeny class factors as 1.5.ae $\times$ 2.5.ae_l 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 $\overline{\F}_{5}$
The base change of $A$ to $\F_{5^{6}}$ is 1.15625.ja 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
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
3.5.a_a_ae$2$3.25.a_a_ja
3.5.a_a_e$2$3.25.a_a_ja
3.5.i_bg_dg$2$3.25.a_a_ja
3.5.e_ab_ay$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.5.a_a_ae$2$3.25.a_a_ja
3.5.a_a_e$2$3.25.a_a_ja
3.5.i_bg_dg$2$3.25.a_a_ja
3.5.e_ab_ay$3$(not in LMFDB)
3.5.ag_m_aq$4$(not in LMFDB)
3.5.ag_y_ack$4$(not in LMFDB)
3.5.ae_i_as$4$(not in LMFDB)
3.5.ac_ae_y$4$(not in LMFDB)
3.5.ac_i_as$4$(not in LMFDB)
3.5.a_a_aw$4$(not in LMFDB)
3.5.a_a_w$4$(not in LMFDB)
3.5.c_ae_ay$4$(not in LMFDB)
3.5.c_i_s$4$(not in LMFDB)
3.5.e_i_s$4$(not in LMFDB)
3.5.g_m_q$4$(not in LMFDB)
3.5.g_y_ck$4$(not in LMFDB)
3.5.am_cl_ahc$6$(not in LMFDB)
3.5.ae_ab_y$6$(not in LMFDB)
3.5.m_cl_hc$6$(not in LMFDB)
3.5.ak_bv_afc$12$(not in LMFDB)
3.5.ai_bj_ads$12$(not in LMFDB)
3.5.ag_p_abc$12$(not in LMFDB)
3.5.ag_bb_acq$12$(not in LMFDB)
3.5.ae_l_ay$12$(not in LMFDB)
3.5.ac_ab_m$12$(not in LMFDB)
3.5.ac_i_as$12$(not in LMFDB)
3.5.ac_l_am$12$(not in LMFDB)
3.5.a_d_aq$12$(not in LMFDB)
3.5.a_d_q$12$(not in LMFDB)
3.5.c_ab_am$12$(not in LMFDB)
3.5.c_l_m$12$(not in LMFDB)
3.5.e_l_y$12$(not in LMFDB)
3.5.g_p_bc$12$(not in LMFDB)
3.5.g_bb_cq$12$(not in LMFDB)
3.5.i_bj_ds$12$(not in LMFDB)
3.5.k_bv_fc$12$(not in LMFDB)
3.5.ae_ad_bg$24$(not in LMFDB)
3.5.ae_n_abg$24$(not in LMFDB)
3.5.ac_ad_q$24$(not in LMFDB)
3.5.ac_n_aq$24$(not in LMFDB)
3.5.c_ad_aq$24$(not in LMFDB)
3.5.c_n_q$24$(not in LMFDB)
3.5.e_ad_abg$24$(not in LMFDB)
3.5.e_n_bg$24$(not in LMFDB)