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Properties

Label 2.5.2.10a
Base 2.1.1.0a1.1
Degree \(10\)
e \(2\)
f \(5\)
c \(10\)

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Defining polynomial over unramified subextension

$x^{2} + 2 a_{1} x + 4 c_{2} + 2$

Invariants

Residue field characteristic: $2$
Degree: $10$
Base field: $\Q_{2}$
Ramification index $e$: $2$
Residue field degree $f$: $5$
Discriminant exponent $c$: $10$
Artin slopes: $[2]$
Swan slopes: $[1]$
Means: $\langle\frac{1}{2}\rangle$
Rams: $(1)$
Field count: $14$ (complete)
Ambiguity: $10$
Mass: $31$
Absolute Mass: $31/5$

Diagrams

Varying

Indices of inseparability: $[1,0]$
Associated inertia: $[1]$
Jump Set: $[1,2]$ (show 1), $[1,3]$ (show 12), $[1,4]$ (show 1)

Galois groups and Hidden Artin slopes

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Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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Showing all 14

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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
2.5.2.10a1.1 2 $( x^{5} + x^{2} + 1 )^{2} + 2 ( x^{5} + x^{2} + 1 ) + 2$ $C_{10}$ (as 10T1) $10$ $10$ $[2]^{5}$ $[1]^{5}$ $[\ ]$ $[\ ]$ $[1, 0]$ $[1]$ $z + t^4$ $[1, 2]$
2.5.2.10a1.2 2 $( x^{5} + x^{2} + 1 )^{2} + 2 ( x^{5} + x^{2} + 1 ) + 6$ $C_{10}$ (as 10T1) $10$ $10$ $[2]^{5}$ $[1]^{5}$ $[\ ]$ $[\ ]$ $[1, 0]$ $[1]$ $z + t^4$ $[1, 4]$
2.5.2.10a2.1 1 $( x^{5} + x^{2} + 1 )^{2} + 2 x ( x^{5} + x^{2} + 1 ) + 2$ $C_2^4 : C_5$ (as 10T8) $80$ $2$ $[2, 2, 2, 2]^{5}$ $[1,1,1,1]^{5}$ $[2,2,2]$ $[1,1,1]$ $[1, 0]$ $[1]$ $z + (t^2 + 1)$ $[1, 3]$
2.5.2.10a2.2 1 $( x^{5} + x^{2} + 1 )^{2} + 2 x ( x^{5} + x^{2} + 1 ) + 4 x^{2} + 2$ $C_2 \times (C_2^4 : C_5)$ (as 10T14) $160$ $2$ $[2, 2, 2, 2]^{10}$ $[1,1,1,1]^{10}$ $[2,2,2]^{2}$ $[1,1,1]^{2}$ $[1, 0]$ $[1]$ $z + (t^2 + 1)$ $[1, 3]$
2.5.2.10a3.1 2 $( x^{5} + x^{2} + 1 )^{2} + \left(2 x^{2} + 2\right) ( x^{5} + x^{2} + 1 ) + 2$ $C_2 \times (C_2^4 : C_5)$ (as 10T14) $160$ $2$ $[2, 2, 2, 2, 2]^{5}$ $[1,1,1,1,1]^{5}$ $[2,2,2,2]$ $[1,1,1,1]$ $[1, 0]$ $[1]$ $z + (t^4 + t^3 + t)$ $[1, 3]$
2.5.2.10a3.2 2 $( x^{5} + x^{2} + 1 )^{2} + \left(2 x^{2} + 2\right) ( x^{5} + x^{2} + 1 ) + 6$ $C_2 \times (C_2^4 : C_5)$ (as 10T14) $160$ $2$ $[2, 2, 2, 2, 2]^{5}$ $[1,1,1,1,1]^{5}$ $[2,2,2,2]$ $[1,1,1,1]$ $[1, 0]$ $[1]$ $z + (t^4 + t^3 + t)$ $[1, 3]$
2.5.2.10a4.1 2 $( x^{5} + x^{2} + 1 )^{2} + \left(2 x^{2} + 2 x + 2\right) ( x^{5} + x^{2} + 1 ) + 2$ $C_2 \times (C_2^4 : C_5)$ (as 10T14) $160$ $2$ $[2, 2, 2, 2, 2]^{5}$ $[1,1,1,1,1]^{5}$ $[2,2,2,2]$ $[1,1,1,1]$ $[1, 0]$ $[1]$ $z + (t^4 + t^3 + t^2 + t + 1)$ $[1, 3]$
2.5.2.10a4.2 2 $( x^{5} + x^{2} + 1 )^{2} + \left(2 x^{2} + 2 x + 2\right) ( x^{5} + x^{2} + 1 ) + 6$ $C_2 \times (C_2^4 : C_5)$ (as 10T14) $160$ $2$ $[2, 2, 2, 2, 2]^{5}$ $[1,1,1,1,1]^{5}$ $[2,2,2,2]$ $[1,1,1,1]$ $[1, 0]$ $[1]$ $z + (t^4 + t^3 + t^2 + t + 1)$ $[1, 3]$
2.5.2.10a5.1 1 $( x^{5} + x^{2} + 1 )^{2} + 2 x^{3} ( x^{5} + x^{2} + 1 ) + 2$ $C_2^4 : C_5$ (as 10T8) $80$ $2$ $[2, 2, 2, 2]^{5}$ $[1,1,1,1]^{5}$ $[2,2,2]$ $[1,1,1]$ $[1, 0]$ $[1]$ $z + (t^4 + t^2)$ $[1, 3]$
2.5.2.10a5.2 1 $( x^{5} + x^{2} + 1 )^{2} + 2 x^{3} ( x^{5} + x^{2} + 1 ) + 4 x + 2$ $C_2 \times (C_2^4 : C_5)$ (as 10T14) $160$ $2$ $[2, 2, 2, 2]^{10}$ $[1,1,1,1]^{10}$ $[2,2,2]^{2}$ $[1,1,1]^{2}$ $[1, 0]$ $[1]$ $z + (t^4 + t^2)$ $[1, 3]$
2.5.2.10a6.1 2 $( x^{5} + x^{2} + 1 )^{2} + \left(2 x^{4} + 2 x^{2}\right) ( x^{5} + x^{2} + 1 ) + 2$ $C_2 \times (C_2^4 : C_5)$ (as 10T14) $160$ $2$ $[2, 2, 2, 2, 2]^{5}$ $[1,1,1,1,1]^{5}$ $[2,2,2,2]$ $[1,1,1,1]$ $[1, 0]$ $[1]$ $z + (t^2 + t + 1)$ $[1, 3]$
2.5.2.10a6.2 2 $( x^{5} + x^{2} + 1 )^{2} + \left(2 x^{4} + 2 x^{2}\right) ( x^{5} + x^{2} + 1 ) + 6$ $C_2 \times (C_2^4 : C_5)$ (as 10T14) $160$ $2$ $[2, 2, 2, 2, 2]^{5}$ $[1,1,1,1,1]^{5}$ $[2,2,2,2]$ $[1,1,1,1]$ $[1, 0]$ $[1]$ $z + (t^2 + t + 1)$ $[1, 3]$
2.5.2.10a7.1 1 $( x^{5} + x^{2} + 1 )^{2} + \left(2 x^{4} + 2 x^{3} + 2 x^{2} + 2 x + 2\right) ( x^{5} + x^{2} + 1 ) + 2$ $C_2^4 : C_5$ (as 10T8) $80$ $2$ $[2, 2, 2, 2]^{5}$ $[1,1,1,1]^{5}$ $[2,2,2]$ $[1,1,1]$ $[1, 0]$ $[1]$ $z + (t^2 + t)$ $[1, 3]$
2.5.2.10a7.2 1 $( x^{5} + x^{2} + 1 )^{2} + \left(2 x^{4} + 2 x^{3} + 2 x^{2} + 2 x + 2\right) ( x^{5} + x^{2} + 1 ) + 4 x^{2} + 2$ $C_2 \times (C_2^4 : C_5)$ (as 10T14) $160$ $2$ $[2, 2, 2, 2]^{10}$ $[1,1,1,1]^{10}$ $[2,2,2]^{2}$ $[1,1,1]^{2}$ $[1, 0]$ $[1]$ $z + (t^2 + t)$ $[1, 3]$
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