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Properties

Label 2.5.4.30a
Base 2.1.1.0a1.1
Degree \(20\)
e \(4\)
f \(5\)
c \(30\)

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Defining polynomial over unramified subextension

$x^{4} + 2 a_{3} x^{3} + 2 b_{2} x^{2} + 4 c_{4} + 2$

Invariants

Residue field characteristic: $2$
Degree: $20$
Base field: $\Q_{2}$
Ramification index $e$: $4$
Residue field degree $f$: $5$
Discriminant exponent $c$: $30$
Artin slopes: $[2,2]$
Swan slopes: $[1,1]$
Means: $\langle\frac{1}{2},\frac{3}{4}\rangle$
Rams: $(1,1)$
Field count: $393$ (complete)
Ambiguity: $20$
Mass: $992$
Absolute Mass: $992/5$

Diagrams

Varying

Indices of inseparability: $[3,2,0]$ (show 379), $[3,3,0]$ (show 14)
Associated inertia: $[1]$ (show 124), $[2]$ (show 200), $[3]$ (show 69)
Jump Set: $[1,2,7]$ (show 13), $[1,3,6]$ (show 10), $[1,3,7]$ (show 360), $[1,3,8]$ (show 10)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 13

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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
2.5.4.30a8.1 $( x^{5} + x^{2} + 1 )^{4} + 2 ( x^{5} + x^{2} + 1 )^{3} + 2 ( x^{5} + x^{2} + 1 )^{2} + 2$ $C_5\times A_4$ (as 20T14) $60$ $5$ not computed not computed not computed not computed $[3, 2, 0]$ $[3]$ $z^3 + (t^3 + t^2 + 1) z + (t^3 + t^2 + t)$ $[1, 2, 7]$
2.5.4.30a9.1 $( x^{5} + x^{2} + 1 )^{4} + 2 x ( x^{5} + x^{2} + 1 )^{3} + 2 ( x^{5} + x^{2} + 1 )^{2} + 2$ $C_2^8.C_{20}$ (as 20T333) $5120$ $2$ not computed not computed not computed not computed $[3, 2, 0]$ $[2]$ $z^3 + (t^3 + t^2 + 1) z + (t^4 + t^3 + t^2)$ $[1, 2, 7]$
2.5.4.30a9.2 $( x^{5} + x^{2} + 1 )^{4} + 2 x ( x^{5} + x^{2} + 1 )^{3} + 2 ( x^{5} + x^{2} + 1 )^{2} + 4 x^{3} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[3, 2, 0]$ $[2]$ $z^3 + (t^3 + t^2 + 1) z + (t^4 + t^3 + t^2)$ $[1, 2, 7]$
2.5.4.30a10.1 $( x^{5} + x^{2} + 1 )^{4} + \left(2 x^{2} + 2\right) ( x^{5} + x^{2} + 1 )^{3} + 2 ( x^{5} + x^{2} + 1 )^{2} + 2$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[3, 2, 0]$ $[1]$ $z^3 + (t^3 + t^2 + 1) z + (t^4 + t + 1)$ $[1, 2, 7]$
2.5.4.30a10.2 $( x^{5} + x^{2} + 1 )^{4} + \left(2 x^{2} + 2\right) ( x^{5} + x^{2} + 1 )^{3} + 2 ( x^{5} + x^{2} + 1 )^{2} + 4 x + 2$ $C_2^5:C_{10}$ (as 20T74) $320$ $4$ not computed not computed not computed not computed $[3, 2, 0]$ $[1]$ $z^3 + (t^3 + t^2 + 1) z + (t^4 + t + 1)$ $[1, 2, 7]$
2.5.4.30a10.3 $( x^{5} + x^{2} + 1 )^{4} + \left(2 x^{2} + 2\right) ( x^{5} + x^{2} + 1 )^{3} + 2 ( x^{5} + x^{2} + 1 )^{2} + 6$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[3, 2, 0]$ $[1]$ $z^3 + (t^3 + t^2 + 1) z + (t^4 + t + 1)$ $[1, 2, 7]$
2.5.4.30a10.4 $( x^{5} + x^{2} + 1 )^{4} + \left(2 x^{2} + 2\right) ( x^{5} + x^{2} + 1 )^{3} + 2 ( x^{5} + x^{2} + 1 )^{2} + 4 x + 6$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[3, 2, 0]$ $[1]$ $z^3 + (t^3 + t^2 + 1) z + (t^4 + t + 1)$ $[1, 2, 7]$
2.5.4.30a11.1 $( x^{5} + x^{2} + 1 )^{4} + \left(2 x^{2} + 2 x + 2\right) ( x^{5} + x^{2} + 1 )^{3} + 2 ( x^{5} + x^{2} + 1 )^{2} + 2$ $C_2^4:F_{16}$ (as 20T286) $3840$ $1$ not computed not computed not computed not computed $[3, 2, 0]$ $[3]$ $z^3 + (t^3 + t^2 + 1) z + (t^3 + t^2 + t + 1)$ $[1, 2, 7]$
2.5.4.30a12.1 $( x^{5} + x^{2} + 1 )^{4} + 2 x^{3} ( x^{5} + x^{2} + 1 )^{3} + 2 ( x^{5} + x^{2} + 1 )^{2} + 2$ $C_2^{10}.C_{10}$ (as 20T427) $10240$ $2$ not computed not computed not computed not computed $[3, 2, 0]$ $[2]$ $z^3 + (t^3 + t^2 + 1) z + (t^4 + t^3 + t^2 + t + 1)$ $[1, 2, 7]$
2.5.4.30a12.2 $( x^{5} + x^{2} + 1 )^{4} + 2 x^{3} ( x^{5} + x^{2} + 1 )^{3} + 2 ( x^{5} + x^{2} + 1 )^{2} + 6$ $C_2^{10}.C_{10}$ (as 20T427) $10240$ $2$ not computed not computed not computed not computed $[3, 2, 0]$ $[2]$ $z^3 + (t^3 + t^2 + 1) z + (t^4 + t^3 + t^2 + t + 1)$ $[1, 2, 7]$
2.5.4.30a13.1 $( x^{5} + x^{2} + 1 )^{4} + \left(2 x^{4} + 2 x^{2}\right) ( x^{5} + x^{2} + 1 )^{3} + 2 ( x^{5} + x^{2} + 1 )^{2} + 2$ $C_2^2:F_{16}$ (as 20T171) $960$ $1$ not computed not computed not computed not computed $[3, 2, 0]$ $[3]$ $z^3 + (t^3 + t^2 + 1) z + (t^2 + t)$ $[1, 2, 7]$
2.5.4.30a14.1 $( x^{5} + x^{2} + 1 )^{4} + \left(2 x^{4} + 2 x^{3} + 2 x^{2} + 2 x + 2\right) ( x^{5} + x^{2} + 1 )^{3} + 2 ( x^{5} + x^{2} + 1 )^{2} + 2$ $C_2^8.C_{20}$ (as 20T333) $5120$ $2$ not computed not computed not computed not computed $[3, 2, 0]$ $[2]$ $z^3 + (t^3 + t^2 + 1) z + (t^3 + t + 1)$ $[1, 2, 7]$
2.5.4.30a14.2 $( x^{5} + x^{2} + 1 )^{4} + \left(2 x^{4} + 2 x^{3} + 2 x^{2} + 2 x + 2\right) ( x^{5} + x^{2} + 1 )^{3} + 2 ( x^{5} + x^{2} + 1 )^{2} + 4 x + 2$ $C_2^9.C_{10}$ (as 20T329) $5120$ $2$ not computed not computed not computed not computed $[3, 2, 0]$ $[2]$ $z^3 + (t^3 + t^2 + 1) z + (t^3 + t + 1)$ $[1, 2, 7]$
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