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Properties

Label 2.1.18.22a1.7
Base \(\Q_{2}\)
Degree \(18\)
e \(18\)
f \(1\)
c \(22\)
Galois group $C_2^2:A_4^2.S_4$ (as 18T588)

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Defining polynomial

Copy content comment:Define the p-adic field
 
Copy content sage:Prec = 100 # Default precision of 100 Q2 = Qp(2, Prec); x = polygen(QQ) K.<a> = Q2.extension(x^18 + 2*x^9 + 2*x^7 + 2*x^5 + 2)
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [2, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1]));
 

\(x^{18} + 2 x^{9} + 2 x^{7} + 2 x^{5} + 2\) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content magma:DefiningPolynomial(K);
 

Invariants

Base field: $\Q_{2}$
Copy content comment:Base field Qp
 
Copy content sage:K.base()
 
Copy content magma:Q2;
 
Degree $d$: $18$
Copy content comment:Degree over Qp
 
Copy content sage:K.absolute_degree()
 
Copy content magma:Degree(K);
 
Ramification index $e$:$18$
Copy content comment:Ramification index
 
Copy content sage:K.absolute_e()
 
Copy content magma:RamificationIndex(K);
 
Residue field degree $f$:$1$
Copy content comment:Residue field degree (Inertia degree)
 
Copy content sage:K.absolute_f()
 
Copy content magma:InertiaDegree(K);
 
Discriminant exponent $c$:$22$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{2}$
Root number: $1$
$\Aut(K/\Q_{2})$: $C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:$[\frac{14}{9}]$
Visible Swan slopes:$[\frac{5}{9}]$
Means:$\langle\frac{5}{18}\rangle$
Rams:$(5)$
Jump set:$[9, 23]$
Roots of unity:$2$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

2.1.3.2a1.1, 2.1.9.8a1.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Canonical tower

Unramified subfield:$\Q_{2}$
Copy content comment:Maximal unramified subextension
 
Copy content sage:K.maximal_unramified_subextension()
 
Relative Eisenstein polynomial: \( x^{18} + 2 x^{9} + 2 x^{7} + 2 x^{5} + 2 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{16} + z^{14} + 1$,$z + 1$
Associated inertia:$6$,$1$
Indices of inseparability:$[5, 0]$

Invariants of the Galois closure

Galois degree: $13824$
Galois group: $C_2^2:A_4^2.S_4$ (as 18T588)
Inertia group: $C_2^8:C_9$ (as 18T368)
Wild inertia group: $C_2^8$
Galois unramified degree: $6$
Galois tame degree: $9$
Galois Artin slopes: $[\frac{4}{3}, \frac{4}{3}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}]$
Galois Swan slopes: $[\frac{1}{3},\frac{1}{3},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9}]$
Galois mean slope: $1.5503472222222223$
Galois splitting model: $x^{18} - 6 x^{17} + 15 x^{16} - 32 x^{15} - 96 x^{14} + 4578 x^{13} - 22652 x^{12} + 45084 x^{11} + 199980 x^{10} - 1249482 x^{9} + 2635932 x^{8} - 822366 x^{7} - 12383532 x^{6} + 8996022 x^{5} + 48404646 x^{4} + 10999530 x^{3} - 134527635 x^{2} - 166997052 x - 149490657$ Copy content Toggle raw display