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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) L.<t> = Q2.extension(x^2 + x + 1) K.<a> = L.extension(x^8 + 8*x^7 + 4*x^6 + (8*t + 8)*x^5 + (8*t + 4)*x^4 + 8*x^3 + 8*t*x^2 + (16*t + 2))
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [19, 144, 472, 1216, 2374, 3720, 4780, 5120, 4599, 3472, 2196, 1152, 494, 168, 44, 8, 1]));
 

$( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{2} + 16 x + 2$ 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 $d$: $16$
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 $f$:$2$
Copy content comment:Residue field degree (Inertia degree)
 
Copy content sage:K.absolute_f()
 
Copy content magma:InertiaDegree(K);
 
Discriminant exponent $c$:$52$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{2}$
Root number: $1$
$\Aut(K/\Q_{2})$: $C_2^2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:$[3, \frac{7}{2}, 4]$
Visible Swan slopes:$[2,\frac{5}{2},3]$
Means:$\langle1, \frac{7}{4}, \frac{19}{8}\rangle$
Rams:$(2, 3, 5)$
Jump set:$[1, 3, 7, 15]$
Roots of unity:$6 = (2^{ 2 } - 1) \cdot 2$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

$\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-2})$, $\Q_{2}(\sqrt{-2\cdot 5})$, 2.2.2.6a1.1, 2.2.4.20a1.16, 2.2.4.22a1.12, 2.2.4.22a1.2

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

Canonical tower

Unramified subfield:$\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} + x + 1 \) Copy content Toggle raw display
Copy content comment:Maximal unramified subextension
 
Copy content sage:K.maximal_unramified_subextension()
 
Relative Eisenstein polynomial: \( x^{8} + 8 x^{7} + 4 x^{6} + \left(8 t + 8\right) x^{5} + \left(8 t + 4\right) x^{4} + 8 x^{3} + 8 t x^{2} + 16 t + 2 \) $\ \in\Q_{2}(t)[x]$ 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:$[19, 14, 8, 0]$

Invariants of the Galois closure

Galois degree: $32$
Galois group: $C_4^2:C_2$ (as 16T27)
Inertia group: not computed
Wild inertia group: not computed
Galois unramified degree: $2$
Galois tame degree: $1$
Galois Artin slopes: $[2, 3, \frac{7}{2}, 4]$
Galois Swan slopes: $[1,2,\frac{5}{2},3]$
Galois mean slope: $3.375$
Galois splitting model:not computed