Properties

Label 5.2.10.30a
Base 5.1.1.0a1.1
Degree \(20\)
e \(10\)
f \(2\)
c \(30\)

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Defining polynomial over unramified subextension

$x^{10} + 5 b_{7} x^{7} + 5 a_{6} x^{6} + 5 d_{0}$

Invariants

Residue field characteristic: $5$
Degree: $20$
Base field: $\Q_{5}$
Ramification index $e$: $10$
Residue field degree $f$: $2$
Discriminant exponent $c$: $30$
Artin slopes: $[\frac{7}{4}]$
Swan slopes: $[\frac{3}{4}]$
Means: $\langle\frac{3}{5}\rangle$
Rams: $(\frac{3}{2})$
Field count: $324$ (complete)
Ambiguity: $4$
Mass: $600$
Absolute Mass: $300$

Diagrams

Varying

Indices of inseparability: $[6,0]$
Associated inertia: $[1,1]$ (show 166), $[1,2]$ (show 158)
Jump Set: undefined

Fields


Showing 1-50 of 324

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Label Packet size Polynomial Galois group Galois degree $\#\Aut(K/\Q_p)$ Artin slope content Swan slope content Hidden Artin slopes Hidden Swan slopes Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
5.2.10.30a1.1 $( x^{2} + 4 x + 2 )^{10} + 20 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.2 $( x^{2} + 4 x + 2 )^{10} + 10 x ( x^{2} + 4 x + 2 )^{7} + 20 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^4:C_8$ (as 20T297) $5000$ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.3 $( x^{2} + 4 x + 2 )^{10} + 15 x ( x^{2} + 4 x + 2 )^{7} + 20 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^4:C_8$ (as 20T297) $5000$ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.4 $( x^{2} + 4 x + 2 )^{10} + 5 ( x^{2} + 4 x + 2 )^{7} + 20 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.5 $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 5\right) ( x^{2} + 4 x + 2 )^{7} + 20 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.6 $( x^{2} + 4 x + 2 )^{10} + \left(15 x + 5\right) ( x^{2} + 4 x + 2 )^{7} + 20 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.7 $( x^{2} + 4 x + 2 )^{10} + 10 ( x^{2} + 4 x + 2 )^{7} + 20 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.8 $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 10\right) ( x^{2} + 4 x + 2 )^{7} + 20 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^4:C_8$ (as 20T297) $5000$ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.9 $( x^{2} + 4 x + 2 )^{10} + 15 ( x^{2} + 4 x + 2 )^{7} + 20 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^4:C_8$ (as 20T297) $5000$ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.10 $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 15\right) ( x^{2} + 4 x + 2 )^{7} + 20 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^4:C_8$ (as 20T297) $5000$ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.11 $( x^{2} + 4 x + 2 )^{10} + 20 ( x^{2} + 4 x + 2 )^{7} + 20 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^4:C_8$ (as 20T297) $5000$ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.12 $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 20\right) ( x^{2} + 4 x + 2 )^{7} + 20 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.13 $( x^{2} + 4 x + 2 )^{10} + \left(15 x + 20\right) ( x^{2} + 4 x + 2 )^{7} + 20 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.14 $( x^{2} + 4 x + 2 )^{10} + 20 ( x^{2} + 4 x + 2 )^{6} + 5$ $C_2\times F_5$ (as 20T13) $40$ $4$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.15 $( x^{2} + 4 x + 2 )^{10} + 5 x ( x^{2} + 4 x + 2 )^{7} + 20 ( x^{2} + 4 x + 2 )^{6} + 5$ $C_5^3:(C_2\times C_4)$ (as 20T183) $1000$ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.16 $( x^{2} + 4 x + 2 )^{10} + 10 x ( x^{2} + 4 x + 2 )^{7} + 20 ( x^{2} + 4 x + 2 )^{6} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.17 $( x^{2} + 4 x + 2 )^{10} + 5 ( x^{2} + 4 x + 2 )^{7} + 20 ( x^{2} + 4 x + 2 )^{6} + 5$ $D_5:F_5$ (as 20T54) $200$ $2$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.18 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 5\right) ( x^{2} + 4 x + 2 )^{7} + 20 ( x^{2} + 4 x + 2 )^{6} + 5$ $C_5^3:(C_2\times C_4)$ (as 20T183) $1000$ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.19 $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 5\right) ( x^{2} + 4 x + 2 )^{7} + 20 ( x^{2} + 4 x + 2 )^{6} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.20 $( x^{2} + 4 x + 2 )^{10} + 10 ( x^{2} + 4 x + 2 )^{7} + 20 ( x^{2} + 4 x + 2 )^{6} + 5$ $D_5:F_5$ (as 20T54) $200$ $2$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.21 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 )^{7} + 20 ( x^{2} + 4 x + 2 )^{6} + 5$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a1.22 $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 20\right) ( x^{2} + 4 x + 2 )^{7} + 20 ( x^{2} + 4 x + 2 )^{6} + 5$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 2$ undefined
5.2.10.30a2.1 $( x^{2} + 4 x + 2 )^{10} + 15 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^4:C_8$ (as 20T297) $5000$ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.2 $( x^{2} + 4 x + 2 )^{10} + 5 x ( x^{2} + 4 x + 2 )^{7} + 15 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.3 $( x^{2} + 4 x + 2 )^{10} + 15 x ( x^{2} + 4 x + 2 )^{7} + 15 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^4:C_8$ (as 20T297) $5000$ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.4 $( x^{2} + 4 x + 2 )^{10} + 5 ( x^{2} + 4 x + 2 )^{7} + 15 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.5 $( x^{2} + 4 x + 2 )^{10} + \left(15 x + 5\right) ( x^{2} + 4 x + 2 )^{7} + 15 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.6 $( x^{2} + 4 x + 2 )^{10} + 10 ( x^{2} + 4 x + 2 )^{7} + 15 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^4:C_8$ (as 20T297) $5000$ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.7 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 )^{7} + 15 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^4:C_8$ (as 20T297) $5000$ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.8 $( x^{2} + 4 x + 2 )^{10} + \left(15 x + 10\right) ( x^{2} + 4 x + 2 )^{7} + 15 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.9 $( x^{2} + 4 x + 2 )^{10} + 15 ( x^{2} + 4 x + 2 )^{7} + 15 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.10 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 15\right) ( x^{2} + 4 x + 2 )^{7} + 15 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.11 $( x^{2} + 4 x + 2 )^{10} + \left(15 x + 15\right) ( x^{2} + 4 x + 2 )^{7} + 15 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.12 $( x^{2} + 4 x + 2 )^{10} + 20 ( x^{2} + 4 x + 2 )^{7} + 15 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.13 $( x^{2} + 4 x + 2 )^{10} + \left(15 x + 20\right) ( x^{2} + 4 x + 2 )^{7} + 15 x ( x^{2} + 4 x + 2 )^{6} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.14 $( x^{2} + 4 x + 2 )^{10} + 15 ( x^{2} + 4 x + 2 )^{6} + 5$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.15 $( x^{2} + 4 x + 2 )^{10} + 5 x ( x^{2} + 4 x + 2 )^{7} + 15 ( x^{2} + 4 x + 2 )^{6} + 5$ $C_5^3:(C_2\times C_4)$ (as 20T183) $1000$ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.16 $( x^{2} + 4 x + 2 )^{10} + 10 x ( x^{2} + 4 x + 2 )^{7} + 15 ( x^{2} + 4 x + 2 )^{6} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.17 $( x^{2} + 4 x + 2 )^{10} + 5 ( x^{2} + 4 x + 2 )^{7} + 15 ( x^{2} + 4 x + 2 )^{6} + 5$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.18 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 5\right) ( x^{2} + 4 x + 2 )^{7} + 15 ( x^{2} + 4 x + 2 )^{6} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.19 $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 5\right) ( x^{2} + 4 x + 2 )^{7} + 15 ( x^{2} + 4 x + 2 )^{6} + 5$ $C_5^3:(C_2\times C_4)$ (as 20T183) $1000$ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.20 $( x^{2} + 4 x + 2 )^{10} + 10 ( x^{2} + 4 x + 2 )^{7} + 15 ( x^{2} + 4 x + 2 )^{6} + 5$ $C_{10}:F_5$ (as 20T52) $200$ $2$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.21 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 )^{7} + 15 ( x^{2} + 4 x + 2 )^{6} + 5$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a2.22 $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 20\right) ( x^{2} + 4 x + 2 )^{7} + 15 ( x^{2} + 4 x + 2 )^{6} + 5$ $D_5:F_5$ (as 20T54) $200$ $2$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 4$ undefined
5.2.10.30a3.1 $( x^{2} + 4 x + 2 )^{10} + 10 ( x^{2} + 4 x + 2 )^{6} + 5$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 1$ undefined
5.2.10.30a3.2 $( x^{2} + 4 x + 2 )^{10} + 5 x ( x^{2} + 4 x + 2 )^{7} + 10 ( x^{2} + 4 x + 2 )^{6} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 1$ undefined
5.2.10.30a3.3 $( x^{2} + 4 x + 2 )^{10} + 10 x ( x^{2} + 4 x + 2 )^{7} + 10 ( x^{2} + 4 x + 2 )^{6} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 1$ undefined
5.2.10.30a3.4 $( x^{2} + 4 x + 2 )^{10} + 5 ( x^{2} + 4 x + 2 )^{7} + 10 ( x^{2} + 4 x + 2 )^{6} + 5$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 1$ undefined
5.2.10.30a3.5 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 5\right) ( x^{2} + 4 x + 2 )^{7} + 10 ( x^{2} + 4 x + 2 )^{6} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 1$ undefined
5.2.10.30a3.6 $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 5\right) ( x^{2} + 4 x + 2 )^{7} + 10 ( x^{2} + 4 x + 2 )^{6} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[6, 0]$ $[1, 1]$ $z^5 + 2,2 z^2 + 1$ undefined
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