Properties

Label 3.5.ak_bu_aey
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:  $( 1 - 4 x + 5 x^{2} )( 1 - 6 x + 17 x^{2} - 30 x^{3} + 25 x^{4} )$
  $1 - 10 x + 46 x^{2} - 128 x^{3} + 230 x^{4} - 250 x^{5} + 125 x^{6}$
Frobenius angles:  $\pm0.0512862249088$, $\pm0.147583617650$, $\pm0.384619558242$
Angle rank:  $2$ (numerical)
Jacobians:  $0$
Isomorphism classes:  1

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $14$ $11060$ $1889048$ $232525440$ $29454708094$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-4$ $18$ $122$ $594$ $3016$ $15576$ $78760$ $392546$ $1954562$ $9760818$

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.ag_r 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.afm 2 $\times$ 1.15625.ja. The endomorphism algebra for each factor is:
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.ac_ac_i$2$3.25.ai_q_ba
3.5.c_ac_ai$2$3.25.ai_q_ba
3.5.k_bu_ey$2$3.25.ai_q_ba
3.5.ae_h_ai$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.5.ac_ac_i$2$3.25.ai_q_ba
3.5.c_ac_ai$2$3.25.ai_q_ba
3.5.k_bu_ey$2$3.25.ai_q_ba
3.5.ae_h_ai$3$(not in LMFDB)
3.5.ai_bi_adq$4$(not in LMFDB)
3.5.ae_k_aba$4$(not in LMFDB)
3.5.e_k_ba$4$(not in LMFDB)
3.5.i_bi_dq$4$(not in LMFDB)
3.5.e_h_i$6$(not in LMFDB)
3.5.ae_d_i$12$(not in LMFDB)
3.5.ac_d_e$12$(not in LMFDB)
3.5.ac_h_ae$12$(not in LMFDB)
3.5.c_d_ae$12$(not in LMFDB)
3.5.c_h_e$12$(not in LMFDB)
3.5.e_d_ai$12$(not in LMFDB)
3.5.ai_bd_acu$24$(not in LMFDB)
3.5.ag_v_ace$24$(not in LMFDB)
3.5.ac_f_ay$24$(not in LMFDB)
3.5.a_ad_ai$24$(not in LMFDB)
3.5.a_ad_i$24$(not in LMFDB)
3.5.c_f_y$24$(not in LMFDB)
3.5.g_v_ce$24$(not in LMFDB)
3.5.i_bd_cu$24$(not in LMFDB)