Properties

Label 5.2.10.34a
Base 5.1.1.0a1.1
Degree \(20\)
e \(10\)
f \(2\)
c \(34\)

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Defining polynomial over unramified subextension

$x^{10} + 5 b_{9} x^{9} + 5 a_{8} x^{8} + 5 d_{0} + 25 c_{10}$

Invariants

Residue field characteristic: $5$
Degree: $20$
Base field: $\Q_{5}$
Ramification index $e$: $10$
Residue field degree $f$: $2$
Discriminant exponent $c$: $34$
Artin slopes: $[2]$
Swan slopes: $[1]$
Means: $\langle\frac{4}{5}\rangle$
Rams: $(2)$
Field count: $648$ (complete)
Ambiguity: $20$
Mass: $600$
Absolute Mass: $300$

Diagrams

Varying

Indices of inseparability: $[8,0]$
Associated inertia: $[1,1]$ (show 408), $[1,2]$ (show 84), $[1,4]$ (show 156)
Jump Set: undefined

Fields


Showing 1-50 of 648

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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
5.2.10.34a1.1 $( x^{2} + 4 x + 2 )^{10} + 15 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^4:(C_2\times C_4)$ (as 20T300) $5000$ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.2 $( x^{2} + 4 x + 2 )^{10} + 5 x ( x^{2} + 4 x + 2 )^{9} + 15 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^4:(C_2\times C_4)$ (as 20T300) $5000$ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.3 $( x^{2} + 4 x + 2 )^{10} + 10 x ( x^{2} + 4 x + 2 )^{9} + 15 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.4 $( x^{2} + 4 x + 2 )^{10} + 5 ( x^{2} + 4 x + 2 )^{9} + 15 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.5 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 5\right) ( x^{2} + 4 x + 2 )^{9} + 15 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^4:(C_2\times C_4)$ (as 20T300) $5000$ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.6 $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 5\right) ( x^{2} + 4 x + 2 )^{9} + 15 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.7 $( x^{2} + 4 x + 2 )^{10} + 10 ( x^{2} + 4 x + 2 )^{9} + 15 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^4:(C_2\times C_4)$ (as 20T300) $5000$ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.8 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 )^{9} + 15 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.9 $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 10\right) ( x^{2} + 4 x + 2 )^{9} + 15 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.10 $( x^{2} + 4 x + 2 )^{10} + 15 ( x^{2} + 4 x + 2 )^{9} + 15 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^4:(C_2\times C_4)$ (as 20T300) $5000$ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.11 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 15\right) ( x^{2} + 4 x + 2 )^{9} + 15 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.12 $( x^{2} + 4 x + 2 )^{10} + 20 ( x^{2} + 4 x + 2 )^{9} + 15 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $D_5\times C_5:F_5$ (as 20T180) $1000$ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.13 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 20\right) ( x^{2} + 4 x + 2 )^{9} + 15 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $D_5\times C_5:F_5$ (as 20T180) $1000$ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.14 $( x^{2} + 4 x + 2 )^{10} + 15 ( x^{2} + 4 x + 2 )^{8} + 5$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.15 $( x^{2} + 4 x + 2 )^{10} + 5 x ( x^{2} + 4 x + 2 )^{9} + 15 ( x^{2} + 4 x + 2 )^{8} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.16 $( x^{2} + 4 x + 2 )^{10} + 10 x ( x^{2} + 4 x + 2 )^{9} + 15 ( x^{2} + 4 x + 2 )^{8} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.17 $( x^{2} + 4 x + 2 )^{10} + 5 ( x^{2} + 4 x + 2 )^{9} + 15 ( x^{2} + 4 x + 2 )^{8} + 5$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.18 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 5\right) ( x^{2} + 4 x + 2 )^{9} + 15 ( x^{2} + 4 x + 2 )^{8} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.19 $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 5\right) ( x^{2} + 4 x + 2 )^{9} + 15 ( x^{2} + 4 x + 2 )^{8} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.20 $( x^{2} + 4 x + 2 )^{10} + 10 ( x^{2} + 4 x + 2 )^{9} + 15 ( x^{2} + 4 x + 2 )^{8} + 5$ $D_5:F_5$ (as 20T54) $200$ $2$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.21 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 )^{9} + 15 ( x^{2} + 4 x + 2 )^{8} + 5$ $D_5:F_5$ (as 20T54) $200$ $2$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a1.22 $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 20\right) ( x^{2} + 4 x + 2 )^{9} + 15 ( x^{2} + 4 x + 2 )^{8} + 5$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.34a2.1 $( x^{2} + 4 x + 2 )^{10} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.2 $( x^{2} + 4 x + 2 )^{10} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x + 25$ $C_5\wr C_4$ (as 20T237) $2500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.3 $( x^{2} + 4 x + 2 )^{10} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x + 50$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.4 $( x^{2} + 4 x + 2 )^{10} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x + 75$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.5 $( x^{2} + 4 x + 2 )^{10} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x + 100$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.6 $( x^{2} + 4 x + 2 )^{10} + 5 x ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.7 $( x^{2} + 4 x + 2 )^{10} + 5 x ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x + 25$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.8 $( x^{2} + 4 x + 2 )^{10} + 5 x ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x + 50$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.9 $( x^{2} + 4 x + 2 )^{10} + 10 x ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.10 $( x^{2} + 4 x + 2 )^{10} + 15 x ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.11 $( x^{2} + 4 x + 2 )^{10} + 15 x ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x + 25$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.12 $( x^{2} + 4 x + 2 )^{10} + 15 x ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x + 50$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.13 $( x^{2} + 4 x + 2 )^{10} + 15 x ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x + 75$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.14 $( x^{2} + 4 x + 2 )^{10} + 20 x ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.15 $( x^{2} + 4 x + 2 )^{10} + 20 x ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x + 25$ $C_5\wr C_4$ (as 20T237) $2500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.16 $( x^{2} + 4 x + 2 )^{10} + 5 ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.17 $( x^{2} + 4 x + 2 )^{10} + 5 ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x + 25$ $C_5\wr C_4$ (as 20T237) $2500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.18 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 5\right) ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.19 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 5\right) ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x + 25$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.20 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 5\right) ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x + 50$ $C_5\wr C_4$ (as 20T237) $2500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.21 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 5\right) ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x + 75$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.22 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 5\right) ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x + 100$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.23 $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 5\right) ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.24 $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 5\right) ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x + 25$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.25 $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 5\right) ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x + 50$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.26 $( x^{2} + 4 x + 2 )^{10} + \left(15 x + 5\right) ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.27 $( x^{2} + 4 x + 2 )^{10} + \left(20 x + 5\right) ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.34a2.28 $( x^{2} + 4 x + 2 )^{10} + \left(20 x + 5\right) ( x^{2} + 4 x + 2 )^{9} + 5 x ( x^{2} + 4 x + 2 )^{8} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x + 25$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
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