Properties

Label 2.1.16.66d1.484
Base \(\Q_{2}\)
Degree \(16\)
e \(16\)
f \(1\)
c \(66\)
Galois group $C_2^6.(D_4\times S_4)$ (as 16T1756)

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^12 + 16*x^9 + 4*x^8 + 16*x^7 + 16*x^6 + 8*x^4 + 16*x^3 + 10)
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [10, 0, 0, 16, 8, 0, 16, 16, 4, 16, 0, 0, 8, 0, 0, 0, 1]));
 

\(x^{16} + 8 x^{12} + 16 x^{9} + 4 x^{8} + 16 x^{7} + 16 x^{6} + 8 x^{4} + 16 x^{3} + 10\) 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 $c$:$66$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{2}$
Root number: $-1$
$\Aut(K/\Q_{2})$: $C_1$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:$[3, 4, \frac{55}{12}, \frac{55}{12}]$
Visible Swan slopes:$[2,3,\frac{43}{12},\frac{43}{12}]$
Means:$\langle1, 2, \frac{67}{24}, \frac{51}{16}\rangle$
Rams:$(2, 4, \frac{19}{3}, \frac{19}{3})$
Jump set:$[1, 3, 7, 15, 31]$
Roots of unity:$2$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

$\Q_{2}(\sqrt{2\cdot 5})$, 2.1.4.11a1.19

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: \( x^{16} + 8 x^{12} + 16 x^{9} + 4 x^{8} + 16 x^{7} + 16 x^{6} + 8 x^{4} + 16 x^{3} + 10 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^8 + 1$,$z^4 + 1$,$z + 1$
Associated inertia:$1$,$1$,$1$
Indices of inseparability:$[51, 48, 32, 16, 0]$

Invariants of the Galois closure

Galois degree: $12288$
Galois group: $C_2^6.(D_4\times S_4)$ (as 16T1756)
Inertia group: $C_2^6.(D_4\times A_4)$ (as 16T1655)
Wild inertia group: not computed
Galois unramified degree: $2$
Galois tame degree: $3$
Galois Artin slopes: $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, 4, \frac{49}{12}, \frac{49}{12}, \frac{55}{12}, \frac{55}{12}]$
Galois Swan slopes: $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},3,\frac{37}{12},\frac{37}{12},\frac{43}{12},\frac{43}{12}]$
Galois mean slope: $4.420247395833333$
Galois splitting model: $x^{16} - 96 x^{13} + 1152 x^{11} + 1152 x^{10} - 9792 x^{9} + 1008 x^{8} + 15360 x^{7} + 11520 x^{6} - 92160 x^{5} + 108672 x^{4} + 1050624 x^{3} - 1949184 x^{2} - 2612736 x + 5925312$ Copy content Toggle raw display