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Properties

Label 16.4.41939526489...0201.1
Degree $16$
Signature $[4, 6]$
Discriminant $11^{2}\cdot 277^{8}$
Root discriminant $22.46$
Ramified primes $11, 277$
Class number $1$
Class group Trivial
Galois group $C_2^3.Q_8.C_6$ (as 16T732)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-163, 879, -1630, 1115, -157, 196, -435, 158, 80, -9, -36, -3, 10, -4, -1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^14 - 4*x^13 + 10*x^12 - 3*x^11 - 36*x^10 - 9*x^9 + 80*x^8 + 158*x^7 - 435*x^6 + 196*x^5 - 157*x^4 + 1115*x^3 - 1630*x^2 + 879*x - 163)
 
gp: K = bnfinit(x^16 - x^14 - 4*x^13 + 10*x^12 - 3*x^11 - 36*x^10 - 9*x^9 + 80*x^8 + 158*x^7 - 435*x^6 + 196*x^5 - 157*x^4 + 1115*x^3 - 1630*x^2 + 879*x - 163, 1)
 

Normalized defining polynomial

\( x^{16} - x^{14} - 4 x^{13} + 10 x^{12} - 3 x^{11} - 36 x^{10} - 9 x^{9} + 80 x^{8} + 158 x^{7} - 435 x^{6} + 196 x^{5} - 157 x^{4} + 1115 x^{3} - 1630 x^{2} + 879 x - 163 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4193952648920057290201=11^{2}\cdot 277^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.46$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $11, 277$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1153180867141365929} a^{15} - \frac{376690893223508193}{1153180867141365929} a^{14} - \frac{475055364847843262}{1153180867141365929} a^{13} - \frac{161825283760333579}{1153180867141365929} a^{12} + \frac{470487857410790018}{1153180867141365929} a^{11} - \frac{288046138140925763}{1153180867141365929} a^{10} + \frac{239430390351731959}{1153180867141365929} a^{9} + \frac{190149490351374057}{1153180867141365929} a^{8} - \frac{366914745023733680}{1153180867141365929} a^{7} + \frac{260055927077842366}{1153180867141365929} a^{6} + \frac{259910312345719374}{1153180867141365929} a^{5} - \frac{199189976185356410}{1153180867141365929} a^{4} - \frac{520406427379023998}{1153180867141365929} a^{3} - \frac{464976862047447840}{1153180867141365929} a^{2} - \frac{205484721600051443}{1153180867141365929} a - \frac{458227314607536403}{1153180867141365929}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 24226.3247022 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.Q_8.C_6$ (as 16T732):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 30 conjugacy class representatives for $C_2^3.Q_8.C_6$
Character table for $C_2^3.Q_8.C_6$ is not computed

Intermediate fields

4.4.76729.1, 8.8.5887339441.1

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

Sibling fields

Degree 16 sibling: data not computed
Degree 32 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$11$11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.12.0.1$x^{12} - x + 7$$1$$12$$0$$C_{12}$$[\ ]^{12}$
277Data not computed