Properties

Label 4.5.am_ct_akj_bbk
Base field $\F_{5}$
Dimension $4$
$p$-rank $4$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

Downloads

Learn more

Invariants

Base field:  $\F_{5}$
Dimension:  $4$
L-polynomial:  $( 1 - 4 x + 5 x^{2} )( 1 - 8 x + 34 x^{2} - 93 x^{3} + 170 x^{4} - 200 x^{5} + 125 x^{6} )$
  $1 - 12 x + 71 x^{2} - 269 x^{3} + 712 x^{4} - 1345 x^{5} + 1775 x^{6} - 1500 x^{7} + 625 x^{8}$
Frobenius angles:  $\pm0.101435245160$, $\pm0.147583617650$, $\pm0.306436956418$, $\pm0.413672014132$
Angle rank:  $4$ (numerical)

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $4$
Slopes:  $[0, 0, 0, 0, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $58$ $365980$ $289808194$ $156569171840$ $94128265961138$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-6$ $24$ $147$ $640$ $3084$ $15711$ $79052$ $393232$ $1957476$ $9772784$

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 $\times$ 3.5.ai_bi_adp 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
4.5.ae_h_d_abg$2$(not in LMFDB)
4.5.e_h_ad_abg$2$(not in LMFDB)
4.5.m_ct_kj_bbk$2$(not in LMFDB)

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

TwistExtension degreeCommon base change
4.5.ae_h_d_abg$2$(not in LMFDB)
4.5.e_h_ad_abg$2$(not in LMFDB)
4.5.m_ct_kj_bbk$2$(not in LMFDB)
4.5.ak_cd_aht_ug$4$(not in LMFDB)
4.5.ag_x_acn_fy$4$(not in LMFDB)
4.5.g_x_cn_fy$4$(not in LMFDB)
4.5.k_cd_ht_ug$4$(not in LMFDB)