Properties

Label 2.1.16.60l1.7
Base \(\Q_{2}\)
Degree \(16\)
e \(16\)
f \(1\)
c \(60\)
Galois group $(D_4\times C_2^3).C_2^3$ (as 16T916)

Related objects

Downloads

Learn more

Show commands: Magma / SageMath

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^16 + 8*x^15 + 8*x^14 + 8*x^13 + 18*x^8 + 16*x^7 + 16*x^5 + 4*x^4 + 16*x^3 + 8*x^2 + 18)
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [18, 0, 8, 16, 4, 16, 0, 16, 18, 0, 0, 0, 0, 8, 8, 8, 1]));
 

\(x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 18 x^{8} + 16 x^{7} + 16 x^{5} + 4 x^{4} + 16 x^{3} + 8 x^{2} + 18\) 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 $e$:$16$
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 $c$:$60$
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:$[2, 3, 4, \frac{9}{2}]$
Visible Swan slopes:$[1,2,3,\frac{7}{2}]$
Means:$\langle\frac{1}{2}, \frac{5}{4}, \frac{17}{8}, \frac{45}{16}\rangle$
Rams:$(1, 3, 7, 11)$
Jump set:$[1, 2, 29, 45, 61]$
Roots of unity:$8 = 2^{ 3 }$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

$\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{2})$, $\Q_{2}(\sqrt{-2})$, 2.1.4.8b1.1, 2.1.8.24c1.4

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: \( x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 18 x^{8} + 16 x^{7} + 16 x^{5} + 4 x^{4} + 16 x^{3} + 8 x^{2} + 18 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^8 + 1$,$z^4 + 1$,$z^2 + 1$,$z + 1$
Associated inertia:$1$,$1$,$1$,$1$
Indices of inseparability:$[45, 34, 20, 8, 0]$

Invariants of the Galois closure

Galois degree: $512$
Galois group: $(D_4\times C_2^3).C_2^3$ (as 16T916)
Inertia group: not computed
Wild inertia group: not computed
Galois unramified degree: $2$
Galois tame degree: $1$
Galois Artin slopes: $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]$
Galois Swan slopes: $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]$
Galois mean slope: $4.25$
Galois splitting model:not computed