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Properties

Label 2.1.8.24c1.62
Base \(\Q_{2}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(24\)
Galois group $C_4\times C_2$ (as 8T2)

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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^8 + 8*x^7 + 4*x^6 + 2*x^4 + 8*x^3 + 4*x^2 + 8*x + 30)
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [30, 8, 4, 8, 2, 0, 4, 8, 1]));
 

\(x^{8} + 8 x^{7} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 30\) 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 $d$: $8$
Copy content comment:Degree over Qp
 
Copy content sage:K.absolute_degree()
 
Copy content magma:Degree(K);
 
Ramification index $e$:$8$
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 $c$:$24$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{2}$
Root number: $-1$
$\Aut(K/\Q_{2})$ $=$ $\Gal(K/\Q_{2})$: $C_2\times C_4$
This field is Galois and abelian over $\Q_{2}.$
Visible Artin slopes:$[2, 3, 4]$
Visible Swan slopes:$[1,2,3]$
Means:$\langle\frac{1}{2}, \frac{5}{4}, \frac{17}{8}\rangle$
Rams:$(1, 3, 7)$
Jump set:$[1, 2, 4, 16]$
Roots of unity:$2$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

$\Q_{2}(\sqrt{-5})$, $\Q_{2}(\sqrt{-2\cdot 5})$, $\Q_{2}(\sqrt{2})$, 2.1.4.8b1.6, 2.1.4.11a1.10, 2.1.4.11a1.11

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: \( x^{8} + 8 x^{7} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 30 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^4 + 1$,$z^2 + 1$,$z + 1$
Associated inertia:$1$,$1$,$1$
Indices of inseparability:$[17, 10, 4, 0]$

Invariants of the Galois closure

Galois degree: $8$
Galois group: $C_2\times C_4$ (as 8T2)
Inertia group: $C_2\times C_4$ (as 8T2)
Wild inertia group: $C_2\times C_4$
Galois unramified degree: $1$
Galois tame degree: $1$
Galois Artin slopes: $[2, 3, 4]$
Galois Swan slopes: $[1,2,3]$
Galois mean slope: $3.0$
Galois splitting model:$x^{8} - 8 x^{6} + 20 x^{4} - 16 x^{2} + 1$