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Properties

Label 16.0.48614142345...6513.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{15}\cdot 19^{8}$
Root discriminant $62.08$
Ramified primes $17, 19$
Class number $16$ (GRH)
Class group $[16]$ (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18684, 44442, 67353, 1424, -10539, -11284, 55708, -48148, 30006, -16316, 6442, -2044, 664, -166, 33, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 33*x^14 - 166*x^13 + 664*x^12 - 2044*x^11 + 6442*x^10 - 16316*x^9 + 30006*x^8 - 48148*x^7 + 55708*x^6 - 11284*x^5 - 10539*x^4 + 1424*x^3 + 67353*x^2 + 44442*x + 18684)
 
gp: K = bnfinit(x^16 - 4*x^15 + 33*x^14 - 166*x^13 + 664*x^12 - 2044*x^11 + 6442*x^10 - 16316*x^9 + 30006*x^8 - 48148*x^7 + 55708*x^6 - 11284*x^5 - 10539*x^4 + 1424*x^3 + 67353*x^2 + 44442*x + 18684, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 33 x^{14} - 166 x^{13} + 664 x^{12} - 2044 x^{11} + 6442 x^{10} - 16316 x^{9} + 30006 x^{8} - 48148 x^{7} + 55708 x^{6} - 11284 x^{5} - 10539 x^{4} + 1424 x^{3} + 67353 x^{2} + 44442 x + 18684 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(48614142345328546750712906513=17^{15}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.08$
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:  $17, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{12} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{12} a^{2} + \frac{1}{3} a$, $\frac{1}{12} a^{9} + \frac{1}{4} a^{3} - \frac{1}{3} a$, $\frac{1}{12} a^{10} - \frac{1}{4} a^{4} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{8} + \frac{1}{12} a^{7} - \frac{1}{6} a^{6} - \frac{1}{24} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{7}{24} a^{2} + \frac{1}{12} a - \frac{1}{2}$, $\frac{1}{24} a^{12} - \frac{1}{24} a^{9} + \frac{1}{8} a^{6} - \frac{1}{6} a^{4} - \frac{1}{8} a^{3} + \frac{1}{6} a$, $\frac{1}{1872} a^{13} + \frac{23}{1872} a^{12} - \frac{25}{1872} a^{11} - \frac{5}{208} a^{10} + \frac{23}{1872} a^{9} + \frac{71}{1872} a^{8} - \frac{319}{1872} a^{7} - \frac{151}{1872} a^{6} - \frac{203}{1872} a^{5} + \frac{17}{208} a^{4} - \frac{145}{624} a^{3} + \frac{389}{1872} a^{2} + \frac{11}{24} a - \frac{11}{52}$, $\frac{1}{7488} a^{14} - \frac{43}{3744} a^{12} - \frac{47}{3744} a^{11} - \frac{95}{3744} a^{10} + \frac{83}{3744} a^{9} - \frac{59}{1872} a^{8} - \frac{463}{3744} a^{7} + \frac{69}{416} a^{6} - \frac{709}{3744} a^{5} - \frac{295}{1248} a^{4} - \frac{1823}{3744} a^{3} + \frac{3299}{7488} a^{2} - \frac{599}{1248} a - \frac{7}{208}$, $\frac{1}{406737364782206427339816576} a^{15} + \frac{5418272308314582280105}{135579121594068809113272192} a^{14} - \frac{38940601263128227792867}{203368682391103213669908288} a^{13} + \frac{133376935145627920635191}{25421085298887901708738536} a^{12} + \frac{37230028286647066017451}{3177635662360987713592317} a^{11} - \frac{191607491893926641713493}{101684341195551606834954144} a^{10} - \frac{4612679690306999410597333}{203368682391103213669908288} a^{9} + \frac{1202626290286234705869179}{203368682391103213669908288} a^{8} - \frac{460501735447231893725995}{8473695099629300569579512} a^{7} - \frac{14743125589919458122909707}{101684341195551606834954144} a^{6} - \frac{3169974894550014264710}{19985129951955897569763} a^{5} + \frac{12178174857008006082841001}{101684341195551606834954144} a^{4} - \frac{162845726020538112719308039}{406737364782206427339816576} a^{3} + \frac{16387768864488525154201911}{45193040531356269704424064} a^{2} - \frac{23205588953182494556445791}{67789560797034404556636096} a - \frac{29537366049052890976985}{65307862039532181653792}$

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

Class group and class number

$C_{16}$, which has order $16$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 95524847.8008 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{-323}) \), 4.0.1773593.1, 8.0.53475746204033.1

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

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R R $16$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
17Data not computed
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$