Properties

Label 5.2.2.2a1.1-1.5.8a
Base 5.2.2.2a1.1
Degree \(5\)
e \(5\)
f \(1\)
c \(8\)

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Defining polynomial

$x^{5} + a_{4} \pi x^{4} + c_{5} \pi^{2} + \pi$

Invariants

Residue field characteristic: $5$
Degree: $5$
Base field: 5.2.2.2a1.1
Ramification index $e$: $5$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Absolute Artin slopes: $[\frac{3}{2}]$
Swan slopes: $[1]$
Means: $\langle\frac{4}{5}\rangle$
Rams: $(1)$
Field count: $20$ (complete)
Ambiguity: $5$
Mass: $24$
Absolute Mass: $6$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 5 }$ within this relative family, not the relative extension.

Galois group: $F_5$ (show 2), $C_4\times D_5$ (show 2), $C_5:F_5$ (show 2), $C_5\times F_5$ (show 2), $C_{10}.D_{10}$ (show 2), $C_2\times C_5^2:C_8$ (show 5), $C_5^2:C_{20}$ (show 4), unknown (1) (incomplete)
Hidden Artin slopes: not computed (show 18), $[\ ]$ (show 2) (incomplete)
Indices of inseparability: $[4,0]$
Associated inertia: $[1,1]$ (show 10), $[1,2]$ (show 4), $[1,4]$ (show 6)
Jump Set: undefined

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Label Polynomial $/ \Q_p$ Galois group $/ \Q_p$ Galois degree $/ \Q_p$ $\#\Aut(K/\Q_p)$ Artin slope content $/ \Q_p$ Swan slope content $/ \Q_p$ Hidden Artin slopes $/ \Q_p$ Hidden Swan slopes $/ \Q_p$ Ind. of Insep. $/ \Q_p$ Assoc. Inertia $/ \Q_p$ Resid. Poly Jump Set
5.2.10.26a1.1 $( x^{2} + 4 x + 2 )^{10} + 5 x ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_{10}.D_{10}$ (as 20T58) $200$ $2$ not computed not computed not computed not computed $[4, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + 1$ undefined
5.2.10.26a2.1 $( x^{2} + 4 x + 2 )^{10} + 10 ( x^{2} + 4 x + 2 )^{5} + 10 x ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^2:C_{20}$ (as 20T126) $500$ $5$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.26a2.2 $( x^{2} + 4 x + 2 )^{10} + 5 ( x^{2} + 4 x + 2 )^{5} + 10 x ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5:F_5$ (as 20T27) $100$ $10$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.26a2.3 $( x^{2} + 4 x + 2 )^{10} + 20 ( x^{2} + 4 x + 2 )^{5} + 10 x ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^2:C_{20}$ (as 20T126) $500$ $5$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + 2$ undefined
5.2.10.26a5.1 $( x^{2} + 4 x + 2 )^{10} + \left(5 x + 15\right) ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[4, 0]$ $[1, 4]$ $z^5 + 2,2 z^4 + t$ undefined
5.2.10.26a6.1 $( x^{2} + 4 x + 2 )^{10} + \left(15 x + 15\right) ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_2\times C_5^2:C_8$ (as 20T113) $400$ $2$ not computed not computed not computed not computed $[4, 0]$ $[1, 4]$ $z^5 + 2,2 z^4 + (t + 2)$ undefined
5.2.10.26a7.1 $( x^{2} + 4 x + 2 )^{10} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $F_5$ (as 20T5) $20$ $20$ $[\frac{3}{2}]_{2}^{2}$ $[\frac{1}{2}]_{2}^{2}$ $[\ ]$ $[\ ]$ $[4, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + (t + 3)$ undefined
5.2.10.26a7.2 $( x^{2} + 4 x + 2 )^{10} + 5 ( x^{2} + 4 x + 2 )^{5} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5\times F_5$ (as 20T29) $100$ $5$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + (t + 3)$ undefined
5.2.10.26a9.1 $( x^{2} + 4 x + 2 )^{10} + \left(15 x + 5\right) ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_4\times D_5$ (as 20T6) $40$ $4$ not computed not computed not computed not computed $[4, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + (2 t + 1)$ undefined
5.2.10.26a10.1 $( x^{2} + 4 x + 2 )^{10} + \left(20 x + 5\right) ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_{10}.D_{10}$ (as 20T58) $200$ $2$ not computed not computed not computed not computed $[4, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + (2 t + 2)$ undefined
5.2.10.26a12.1 $( x^{2} + 4 x + 2 )^{10} + \left(20 x + 20\right) ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_2\times C_5^2:C_8$ (as 20T113) $400$ $2$ not computed not computed not computed not computed $[4, 0]$ $[1, 4]$ $z^5 + 2,2 z^4 + (3 t + 1)$ undefined
5.2.10.26a13.1 $( x^{2} + 4 x + 2 )^{10} + 20 ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_2\times C_5^2:C_8$ (as 20T113) $400$ $2$ not computed not computed not computed not computed $[4, 0]$ $[1, 4]$ $z^5 + 2,2 z^4 + (3 t + 2)$ undefined
5.2.10.26a14.1 $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 20\right) ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_4\times D_5$ (as 20T6) $40$ $4$ not computed not computed not computed not computed $[4, 0]$ $[1, 2]$ $z^5 + 2,2 z^4 + (3 t + 4)$ undefined
5.2.10.26a16.1 $( x^{2} + 4 x + 2 )^{10} + 10 ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_2\times C_5^2:C_8$ (as 20T113) $400$ $2$ not computed not computed not computed not computed $[4, 0]$ $[1, 4]$ $z^5 + 2,2 z^4 + (4 t + 1)$ undefined
5.2.10.26a17.1 $( x^{2} + 4 x + 2 )^{10} + 10 ( x^{2} + 4 x + 2 )^{5} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5\times F_5$ (as 20T29) $100$ $5$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + (4 t + 2)$ undefined
5.2.10.26a17.2 $( x^{2} + 4 x + 2 )^{10} + 5 ( x^{2} + 4 x + 2 )^{5} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $F_5$ (as 20T5) $20$ $20$ $[\frac{3}{2}]_{2}^{2}$ $[\frac{1}{2}]_{2}^{2}$ $[\ ]$ $[\ ]$ $[4, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + (4 t + 2)$ undefined
5.2.10.26a18.1 $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 10\right) ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_2\times C_5^2:C_8$ (as 20T113) $400$ $2$ not computed not computed not computed not computed $[4, 0]$ $[1, 4]$ $z^5 + 2,2 z^4 + (4 t + 3)$ undefined
5.2.10.26a19.1 $( x^{2} + 4 x + 2 )^{10} + 15 x ( x^{2} + 4 x + 2 )^{5} + \left(15 x + 10\right) ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^2:C_{20}$ (as 20T126) $500$ $5$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + (4 t + 4)$ undefined
5.2.10.26a19.2 $( x^{2} + 4 x + 2 )^{10} + 5 x ( x^{2} + 4 x + 2 )^{5} + \left(15 x + 10\right) ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5^2:C_{20}$ (as 20T126) $500$ $5$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + (4 t + 4)$ undefined
5.2.10.26a19.3 $( x^{2} + 4 x + 2 )^{10} + 20 x ( x^{2} + 4 x + 2 )^{5} + \left(15 x + 10\right) ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ $C_5:F_5$ (as 20T27) $100$ $10$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^5 + 2,2 z^4 + (4 t + 4)$ undefined
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