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Properties

Label 2.1.10.16a
Base 2.1.1.0a1.1
Degree \(10\)
e \(10\)
f \(1\)
c \(16\)

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Defining polynomial

$x^{10} + 2 b_{9} x^{9} + 2 a_{7} x^{7} + 4 c_{14} x^{4} + 4 b_{13} x^{3} + 4 b_{11} x + 2$

Invariants

Residue field characteristic: $2$
Degree: $10$
Base field: $\Q_{2}$
Ramification index $e$: $10$
Residue field degree $f$: $1$
Discriminant exponent $c$: $16$
Artin slopes: $[\frac{12}{5}]$
Swan slopes: $[\frac{7}{5}]$
Means: $\langle\frac{7}{10}\rangle$
Rams: $(7)$
Field count: $16$ (complete)
Ambiguity: $2$
Mass: $8$
Absolute Mass: $8$

Diagrams

Varying

Indices of inseparability: $[7,0]$
Associated inertia: $[4,1]$
Jump Set: $[5,15]$

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 16

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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
2.1.10.16a1.1 4 $x^{10} + 2 x^{7} + 2$ $(C_2^4 : C_5):C_4$ (as 10T25) $320$ $2$ $[\frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$ $[\frac{7}{5},\frac{7}{5},\frac{7}{5},\frac{7}{5}]_{5}^{4}$ $[\frac{12}{5},\frac{12}{5},\frac{12}{5}]^{4}$ $[\frac{7}{5},\frac{7}{5},\frac{7}{5}]^{4}$ $[7, 0]$ $[4, 1]$ $z^8 + z^6 + 1,z + 1$ $[5, 15]$
2.1.10.16a1.2 4 $x^{10} + 2 x^{7} + 4 x^{4} + 2$ $(C_2^4 : C_5):C_4$ (as 10T24) $320$ $2$ $[\frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$ $[\frac{7}{5},\frac{7}{5},\frac{7}{5},\frac{7}{5}]_{5}^{4}$ $[\frac{12}{5},\frac{12}{5},\frac{12}{5}]^{4}$ $[\frac{7}{5},\frac{7}{5},\frac{7}{5}]^{4}$ $[7, 0]$ $[4, 1]$ $z^8 + z^6 + 1,z + 1$ $[5, 15]$
2.1.10.16a1.3 4 $x^{10} + 2 x^{7} + 4 x^{3} + 2$ $(C_2^4 : C_5):C_4$ (as 10T25) $320$ $2$ $[\frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$ $[\frac{7}{5},\frac{7}{5},\frac{7}{5},\frac{7}{5}]_{5}^{4}$ $[\frac{12}{5},\frac{12}{5},\frac{12}{5}]^{4}$ $[\frac{7}{5},\frac{7}{5},\frac{7}{5}]^{4}$ $[7, 0]$ $[4, 1]$ $z^8 + z^6 + 1,z + 1$ $[5, 15]$
2.1.10.16a1.4 4 $x^{10} + 2 x^{7} + 4 x^{4} + 4 x^{3} + 2$ $(C_2^4 : C_5):C_4$ (as 10T24) $320$ $2$ $[\frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$ $[\frac{7}{5},\frac{7}{5},\frac{7}{5},\frac{7}{5}]_{5}^{4}$ $[\frac{12}{5},\frac{12}{5},\frac{12}{5}]^{4}$ $[\frac{7}{5},\frac{7}{5},\frac{7}{5}]^{4}$ $[7, 0]$ $[4, 1]$ $z^8 + z^6 + 1,z + 1$ $[5, 15]$
2.1.10.16a1.5 4 $x^{10} + 2 x^{7} + 4 x + 2$ $(C_2^4 : C_5):C_4$ (as 10T25) $320$ $2$ $[\frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$ $[\frac{7}{5},\frac{7}{5},\frac{7}{5},\frac{7}{5}]_{5}^{4}$ $[\frac{12}{5},\frac{12}{5},\frac{12}{5}]^{4}$ $[\frac{7}{5},\frac{7}{5},\frac{7}{5}]^{4}$ $[7, 0]$ $[4, 1]$ $z^8 + z^6 + 1,z + 1$ $[5, 15]$
2.1.10.16a1.6 4 $x^{10} + 2 x^{7} + 4 x^{4} + 4 x + 2$ $(C_2^4 : C_5):C_4$ (as 10T24) $320$ $2$ $[\frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$ $[\frac{7}{5},\frac{7}{5},\frac{7}{5},\frac{7}{5}]_{5}^{4}$ $[\frac{12}{5},\frac{12}{5},\frac{12}{5}]^{4}$ $[\frac{7}{5},\frac{7}{5},\frac{7}{5}]^{4}$ $[7, 0]$ $[4, 1]$ $z^8 + z^6 + 1,z + 1$ $[5, 15]$
2.1.10.16a1.7 4 $x^{10} + 2 x^{7} + 4 x^{3} + 4 x + 2$ $(C_2^4 : C_5):C_4$ (as 10T25) $320$ $2$ $[\frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$ $[\frac{7}{5},\frac{7}{5},\frac{7}{5},\frac{7}{5}]_{5}^{4}$ $[\frac{12}{5},\frac{12}{5},\frac{12}{5}]^{4}$ $[\frac{7}{5},\frac{7}{5},\frac{7}{5}]^{4}$ $[7, 0]$ $[4, 1]$ $z^8 + z^6 + 1,z + 1$ $[5, 15]$
2.1.10.16a1.8 4 $x^{10} + 2 x^{7} + 4 x^{4} + 4 x^{3} + 4 x + 2$ $(C_2^4 : C_5):C_4$ (as 10T24) $320$ $2$ $[\frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$ $[\frac{7}{5},\frac{7}{5},\frac{7}{5},\frac{7}{5}]_{5}^{4}$ $[\frac{12}{5},\frac{12}{5},\frac{12}{5}]^{4}$ $[\frac{7}{5},\frac{7}{5},\frac{7}{5}]^{4}$ $[7, 0]$ $[4, 1]$ $z^8 + z^6 + 1,z + 1$ $[5, 15]$
2.1.10.16a1.9 8 $x^{10} + 2 x^{9} + 2 x^{7} + 2$ $((C_2^4 : C_5):C_4)\times C_2$ (as 10T29) $640$ $2$ $[2, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$ $[1,\frac{7}{5},\frac{7}{5},\frac{7}{5},\frac{7}{5}]_{5}^{4}$ $[2,\frac{12}{5},\frac{12}{5},\frac{12}{5}]^{4}$ $[1,\frac{7}{5},\frac{7}{5},\frac{7}{5}]^{4}$ $[7, 0]$ $[4, 1]$ $z^8 + z^6 + 1,z + 1$ $[5, 15]$
2.1.10.16a1.10 8 $x^{10} + 2 x^{9} + 2 x^{7} + 4 x^{4} + 2$ $((C_2^4 : C_5):C_4)\times C_2$ (as 10T29) $640$ $2$ $[2, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$ $[1,\frac{7}{5},\frac{7}{5},\frac{7}{5},\frac{7}{5}]_{5}^{4}$ $[2,\frac{12}{5},\frac{12}{5},\frac{12}{5}]^{4}$ $[1,\frac{7}{5},\frac{7}{5},\frac{7}{5}]^{4}$ $[7, 0]$ $[4, 1]$ $z^8 + z^6 + 1,z + 1$ $[5, 15]$
2.1.10.16a1.11 8 $x^{10} + 2 x^{9} + 2 x^{7} + 4 x^{3} + 2$ $((C_2^4 : C_5):C_4)\times C_2$ (as 10T29) $640$ $2$ $[2, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$ $[1,\frac{7}{5},\frac{7}{5},\frac{7}{5},\frac{7}{5}]_{5}^{4}$ $[2,\frac{12}{5},\frac{12}{5},\frac{12}{5}]^{4}$ $[1,\frac{7}{5},\frac{7}{5},\frac{7}{5}]^{4}$ $[7, 0]$ $[4, 1]$ $z^8 + z^6 + 1,z + 1$ $[5, 15]$
2.1.10.16a1.12 8 $x^{10} + 2 x^{9} + 2 x^{7} + 4 x^{4} + 4 x^{3} + 2$ $((C_2^4 : C_5):C_4)\times C_2$ (as 10T29) $640$ $2$ $[2, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$ $[1,\frac{7}{5},\frac{7}{5},\frac{7}{5},\frac{7}{5}]_{5}^{4}$ $[2,\frac{12}{5},\frac{12}{5},\frac{12}{5}]^{4}$ $[1,\frac{7}{5},\frac{7}{5},\frac{7}{5}]^{4}$ $[7, 0]$ $[4, 1]$ $z^8 + z^6 + 1,z + 1$ $[5, 15]$
2.1.10.16a1.13 8 $x^{10} + 2 x^{9} + 2 x^{7} + 4 x + 2$ $((C_2^4 : C_5):C_4)\times C_2$ (as 10T29) $640$ $2$ $[2, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$ $[1,\frac{7}{5},\frac{7}{5},\frac{7}{5},\frac{7}{5}]_{5}^{4}$ $[2,\frac{12}{5},\frac{12}{5},\frac{12}{5}]^{4}$ $[1,\frac{7}{5},\frac{7}{5},\frac{7}{5}]^{4}$ $[7, 0]$ $[4, 1]$ $z^8 + z^6 + 1,z + 1$ $[5, 15]$
2.1.10.16a1.14 8 $x^{10} + 2 x^{9} + 2 x^{7} + 4 x^{4} + 4 x + 2$ $((C_2^4 : C_5):C_4)\times C_2$ (as 10T29) $640$ $2$ $[2, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$ $[1,\frac{7}{5},\frac{7}{5},\frac{7}{5},\frac{7}{5}]_{5}^{4}$ $[2,\frac{12}{5},\frac{12}{5},\frac{12}{5}]^{4}$ $[1,\frac{7}{5},\frac{7}{5},\frac{7}{5}]^{4}$ $[7, 0]$ $[4, 1]$ $z^8 + z^6 + 1,z + 1$ $[5, 15]$
2.1.10.16a1.15 8 $x^{10} + 2 x^{9} + 2 x^{7} + 4 x^{3} + 4 x + 2$ $((C_2^4 : C_5):C_4)\times C_2$ (as 10T29) $640$ $2$ $[2, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$ $[1,\frac{7}{5},\frac{7}{5},\frac{7}{5},\frac{7}{5}]_{5}^{4}$ $[2,\frac{12}{5},\frac{12}{5},\frac{12}{5}]^{4}$ $[1,\frac{7}{5},\frac{7}{5},\frac{7}{5}]^{4}$ $[7, 0]$ $[4, 1]$ $z^8 + z^6 + 1,z + 1$ $[5, 15]$
2.1.10.16a1.16 8 $x^{10} + 2 x^{9} + 2 x^{7} + 4 x^{4} + 4 x^{3} + 4 x + 2$ $((C_2^4 : C_5):C_4)\times C_2$ (as 10T29) $640$ $2$ $[2, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$ $[1,\frac{7}{5},\frac{7}{5},\frac{7}{5},\frac{7}{5}]_{5}^{4}$ $[2,\frac{12}{5},\frac{12}{5},\frac{12}{5}]^{4}$ $[1,\frac{7}{5},\frac{7}{5},\frac{7}{5}]^{4}$ $[7, 0]$ $[4, 1]$ $z^8 + z^6 + 1,z + 1$ $[5, 15]$
  displayed columns for results