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Properties

Label 16.0.53178385204...2816.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{12}\cdot 13^{12}$
Root discriminant $62.42$
Ramified primes $2, 3, 13$
Class number $2048$ (GRH)
Class group $[2, 2, 4, 4, 4, 8]$ (GRH)
Galois group $Q_{16}$ (as 16T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1385757, -8146152, 16341750, -14175360, 9984474, -5077560, 2450022, -940332, 390370, -121952, 42818, -9704, 2602, -352, 74, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 74*x^14 - 352*x^13 + 2602*x^12 - 9704*x^11 + 42818*x^10 - 121952*x^9 + 390370*x^8 - 940332*x^7 + 2450022*x^6 - 5077560*x^5 + 9984474*x^4 - 14175360*x^3 + 16341750*x^2 - 8146152*x + 1385757)
 
gp: K = bnfinit(x^16 - 4*x^15 + 74*x^14 - 352*x^13 + 2602*x^12 - 9704*x^11 + 42818*x^10 - 121952*x^9 + 390370*x^8 - 940332*x^7 + 2450022*x^6 - 5077560*x^5 + 9984474*x^4 - 14175360*x^3 + 16341750*x^2 - 8146152*x + 1385757, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 74 x^{14} - 352 x^{13} + 2602 x^{12} - 9704 x^{11} + 42818 x^{10} - 121952 x^{9} + 390370 x^{8} - 940332 x^{7} + 2450022 x^{6} - 5077560 x^{5} + 9984474 x^{4} - 14175360 x^{3} + 16341750 x^{2} - 8146152 x + 1385757 \)

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:  \(53178385204239177923287842816=2^{32}\cdot 3^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.42$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{26} a^{8} - \frac{1}{13} a^{7} - \frac{2}{13} a^{6} - \frac{1}{13} a^{5} + \frac{1}{13} a^{4} + \frac{3}{13} a^{3} - \frac{5}{13} a^{2} + \frac{1}{13} a + \frac{3}{26}$, $\frac{1}{26} a^{9} - \frac{4}{13} a^{7} - \frac{5}{13} a^{6} - \frac{1}{13} a^{5} + \frac{5}{13} a^{4} + \frac{1}{13} a^{3} + \frac{4}{13} a^{2} + \frac{7}{26} a + \frac{3}{13}$, $\frac{1}{26} a^{10} - \frac{4}{13} a^{6} - \frac{3}{13} a^{5} - \frac{4}{13} a^{4} + \frac{2}{13} a^{3} + \frac{5}{26} a^{2} - \frac{2}{13} a - \frac{1}{13}$, $\frac{1}{26} a^{11} - \frac{4}{13} a^{7} - \frac{3}{13} a^{6} - \frac{4}{13} a^{5} + \frac{2}{13} a^{4} + \frac{5}{26} a^{3} - \frac{2}{13} a^{2} - \frac{1}{13} a$, $\frac{1}{78} a^{12} - \frac{1}{78} a^{11} - \frac{1}{78} a^{10} - \frac{1}{78} a^{9} + \frac{1}{78} a^{8} - \frac{4}{39} a^{7} + \frac{16}{39} a^{6} + \frac{14}{39} a^{5} - \frac{35}{78} a^{4} - \frac{1}{2} a^{3} + \frac{1}{26} a^{2} - \frac{3}{26} a - \frac{1}{26}$, $\frac{1}{78} a^{13} + \frac{1}{78} a^{11} + \frac{1}{78} a^{10} - \frac{1}{78} a^{8} - \frac{2}{13} a^{7} - \frac{1}{13} a^{6} + \frac{17}{78} a^{5} + \frac{2}{39} a^{4} + \frac{9}{26} a^{3} + \frac{5}{26} a^{2} - \frac{3}{13} a + \frac{3}{26}$, $\frac{1}{10218} a^{14} + \frac{29}{10218} a^{13} - \frac{19}{3406} a^{12} + \frac{1}{786} a^{11} - \frac{10}{1703} a^{10} - \frac{7}{3406} a^{9} - \frac{5}{3406} a^{8} - \frac{1133}{5109} a^{7} + \frac{903}{3406} a^{6} + \frac{3697}{10218} a^{5} - \frac{4115}{10218} a^{4} + \frac{373}{3406} a^{3} - \frac{770}{1703} a^{2} - \frac{59}{3406} a + \frac{1497}{3406}$, $\frac{1}{409196745792066865917413327824291458} a^{15} - \frac{9629692159992074727688374683966}{204598372896033432958706663912145729} a^{14} - \frac{7583631751981058325548946183667}{3824268652262307158106666615180294} a^{13} + \frac{2425185130528596931183102686759101}{409196745792066865917413327824291458} a^{12} - \frac{818371724957920997254270573198367}{68199457632011144319568887970715243} a^{11} + \frac{4050831622031377481336096280915}{807094173159895199048152520363494} a^{10} - \frac{3128539406407795599847453850071205}{204598372896033432958706663912145729} a^{9} - \frac{3339998959690157204762102693516750}{204598372896033432958706663912145729} a^{8} - \frac{138688244912311905969194217987803567}{409196745792066865917413327824291458} a^{7} + \frac{33990256788769798218290790440592690}{68199457632011144319568887970715243} a^{6} + \frac{72520012223683642166127191580059377}{409196745792066865917413327824291458} a^{5} + \frac{34078963250166489287797195025478485}{409196745792066865917413327824291458} a^{4} - \frac{22891723168707513709618350091480017}{68199457632011144319568887970715243} a^{3} + \frac{49253398858850673638235865437871569}{136398915264022288639137775941430486} a^{2} + \frac{17899622531087597188491462179093818}{68199457632011144319568887970715243} a + \frac{38035031808914063497250321055225}{637378108710384526351111102530049}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{8}$, which has order $2048$ (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:  \( 29886.0391883 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_{16}$ (as 16T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $Q_{16}$
Character table for $Q_{16}$

Intermediate fields

\(\Q(\sqrt{39}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{3}, \sqrt{13})\), 4.4.8112.1 x2, 4.4.7488.1 x2, 8.8.9475854336.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$13$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$