# 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

## 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.

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

 $r$ 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.
 Twist Extension Degree Common 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)