Properties

Label 3.5.ah_bd_ada
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

Related objects

Downloads

Learn more

Invariants

Base field:  $\F_{5}$
Dimension:  $3$
L-polynomial:  $( 1 - 4 x + 5 x^{2} )( 1 - 2 x + 5 x^{2} )( 1 - x + 5 x^{2} )$
  $1 - 7 x + 29 x^{2} - 78 x^{3} + 145 x^{4} - 175 x^{5} + 125 x^{6}$
Frobenius angles:  $\pm0.147583617650$, $\pm0.352416382350$, $\pm0.428216853436$
Angle rank:  $2$ (numerical)
Jacobians:  $0$
Isomorphism classes:  20

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $40$ $22400$ $2527840$ $243712000$ $29484336200$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $35$ $158$ $623$ $3019$ $15680$ $79183$ $393023$ $1954454$ $9761675$

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 $\F_{5^{4}}$.

Endomorphism algebra over $\F_{5}$
The isogeny class factors as 1.5.ae $\times$ 1.5.ac $\times$ 1.5.ab 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 $\F_{5^{4}}$ is 1.625.abf $\times$ 1.625.o 2 . 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.af_r_abq$2$3.25.j_bn_ew
3.5.ad_j_aw$2$3.25.j_bn_ew
3.5.ab_f_as$2$3.25.j_bn_ew
3.5.b_f_s$2$3.25.j_bn_ew
3.5.d_j_w$2$3.25.j_bn_ew
3.5.f_r_bq$2$3.25.j_bn_ew
3.5.h_bd_da$2$3.25.j_bn_ew

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

TwistExtension degreeCommon base change
3.5.af_r_abq$2$3.25.j_bn_ew
3.5.ad_j_aw$2$3.25.j_bn_ew
3.5.ab_f_as$2$3.25.j_bn_ew
3.5.b_f_s$2$3.25.j_bn_ew
3.5.d_j_w$2$3.25.j_bn_ew
3.5.f_r_bq$2$3.25.j_bn_ew
3.5.h_bd_da$2$3.25.j_bn_ew
3.5.aj_bn_aec$4$(not in LMFDB)
3.5.ah_x_acc$4$(not in LMFDB)
3.5.af_x_acc$4$(not in LMFDB)
3.5.ad_p_aba$4$(not in LMFDB)
3.5.ab_ab_g$4$(not in LMFDB)
3.5.ab_l_ag$4$(not in LMFDB)
3.5.b_ab_ag$4$(not in LMFDB)
3.5.b_l_g$4$(not in LMFDB)
3.5.d_p_ba$4$(not in LMFDB)
3.5.f_x_cc$4$(not in LMFDB)
3.5.h_x_cc$4$(not in LMFDB)
3.5.j_bn_ec$4$(not in LMFDB)
3.5.ab_ad_i$8$(not in LMFDB)
3.5.ab_n_ai$8$(not in LMFDB)
3.5.b_ad_ai$8$(not in LMFDB)
3.5.b_n_i$8$(not in LMFDB)
3.5.af_u_abz$12$(not in LMFDB)
3.5.ad_g_at$12$(not in LMFDB)
3.5.ad_m_abd$12$(not in LMFDB)
3.5.ab_c_av$12$(not in LMFDB)
3.5.b_c_v$12$(not in LMFDB)
3.5.d_g_t$12$(not in LMFDB)
3.5.d_m_bd$12$(not in LMFDB)
3.5.f_u_bz$12$(not in LMFDB)