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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^12 + 2*x^7 + 2)
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1]));
 

\(x^{12} + 2 x^{7} + 2\) 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 $d$: $12$
Copy content comment:Degree over Qp
 
Copy content sage:K.absolute_degree()
 
Copy content magma:Degree(K);
 
Ramification index $e$:$12$
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 $c$:$18$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{2}(\sqrt{5})$
Root number: $1$
$\Aut(K/\Q_{2})$: $C_1$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:$[\frac{16}{9}, \frac{16}{9}]$
Visible Swan slopes:$[\frac{7}{9},\frac{7}{9}]$
Means:$\langle\frac{7}{18}, \frac{7}{12}\rangle$
Rams:$(\frac{7}{3}, \frac{7}{3})$
Jump set:$[3, 9, 19]$
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

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: \( x^{12} + 2 x^{7} + 2 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^8 + z^4 + 1$,$z + 1$
Associated inertia:$2$,$1$
Indices of inseparability:$[7, 7, 0]$

Invariants of the Galois closure

Galois degree: $3456$
Galois group: $C_2^6:C_9:C_6$ (as 12T254)
Inertia group: $C_2^6:C_9$ (as 12T166)
Wild inertia group: $C_2^6$
Galois unramified degree: $6$
Galois tame degree: $9$
Galois Artin slopes: $[\frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}]$
Galois Swan slopes: $[\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]$
Galois mean slope: $1.7638888888888888$
Galois splitting model: $x^{12} - 2 x^{11} - 16 x^{10} + 4 x^{9} + 120 x^{8} + 558 x^{7} - 2632 x^{6} - 1396 x^{5} + 14098 x^{4} - 21312 x^{3} - 28232 x^{2} + 82280 x - 46706$ Copy content Toggle raw display