Properties

Label 4.5.al_ch_ahz_um
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

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Invariants

Base field:  $\F_{5}$
Dimension:  $4$
L-polynomial:  $( 1 - 4 x + 5 x^{2} )( 1 - 7 x + 26 x^{2} - 68 x^{3} + 130 x^{4} - 175 x^{5} + 125 x^{6} )$
  $1 - 11 x + 59 x^{2} - 207 x^{3} + 532 x^{4} - 1035 x^{5} + 1475 x^{6} - 1375 x^{7} + 625 x^{8}$
Frobenius angles:  $\pm0.0605820805461$, $\pm0.147583617650$, $\pm0.287119770935$, $\pm0.511693182811$
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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $64$ $340480$ $236305216$ $145954242560$ $97099396455424$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-5$ $23$ $121$ $599$ $3180$ $15839$ $77849$ $389679$ $1956595$ $9779718$

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.ah_ba_acq 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.ad_d_b_am$2$(not in LMFDB)
4.5.d_d_ab_am$2$(not in LMFDB)
4.5.l_ch_hz_um$2$(not in LMFDB)

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

TwistExtension degreeCommon base change
4.5.ad_d_b_am$2$(not in LMFDB)
4.5.d_d_ab_am$2$(not in LMFDB)
4.5.l_ch_hz_um$2$(not in LMFDB)
4.5.aj_bt_afz_pg$4$(not in LMFDB)
4.5.af_r_abz_eu$4$(not in LMFDB)
4.5.f_r_bz_eu$4$(not in LMFDB)
4.5.j_bt_fz_pg$4$(not in LMFDB)