Properties

Label 4.5.al_cj_ain_wd
Base Field $\F_{5}$
Dimension $4$
Ordinary No
$p$-rank $3$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{5}$
Dimension:  $4$
L-polynomial:  $( 1 - 3 x + 5 x^{2} )( 1 - 8 x + 32 x^{2} - 85 x^{3} + 160 x^{4} - 200 x^{5} + 125 x^{6} )$
Frobenius angles:  $\pm0.0657033817182$, $\pm0.238557099512$, $\pm0.265942140215$, $\pm0.475140873389$
Angle rank:  $4$ (numerical)

This isogeny class is not simple.

Newton polygon

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1/2, 1/2, 1, 1, 1]$

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 75 412425 285123600 156154415625 96915522862875 60075452420942400 36982220554645153275 23089333365449331215625 14518910772898452925516800 9102589711207377594106379625

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -5 27 145 643 3175 15747 77555 387363 1948690 9773827

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
The isogeny class factors as 1.5.ad $\times$ 3.5.ai_bg_adh and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{5}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.5.af_n_abd_cn$2$(not in LMFDB)
4.5.f_n_bd_cn$2$(not in LMFDB)
4.5.l_cj_in_wd$2$(not in LMFDB)