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Properties

Label 16.0.58102542838...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 5^{10}\cdot 53^{6}$
Root discriminant $40.76$
Ramified primes $2, 5, 53$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group 16T1048

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3281, 10270, 15226, 15322, 14900, 16782, 19806, 19394, 12846, 5218, 1574, 582, 148, 34, 18, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 18*x^14 + 34*x^13 + 148*x^12 + 582*x^11 + 1574*x^10 + 5218*x^9 + 12846*x^8 + 19394*x^7 + 19806*x^6 + 16782*x^5 + 14900*x^4 + 15322*x^3 + 15226*x^2 + 10270*x + 3281)
 
gp: K = bnfinit(x^16 - 2*x^15 + 18*x^14 + 34*x^13 + 148*x^12 + 582*x^11 + 1574*x^10 + 5218*x^9 + 12846*x^8 + 19394*x^7 + 19806*x^6 + 16782*x^5 + 14900*x^4 + 15322*x^3 + 15226*x^2 + 10270*x + 3281, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 18 x^{14} + 34 x^{13} + 148 x^{12} + 582 x^{11} + 1574 x^{10} + 5218 x^{9} + 12846 x^{8} + 19394 x^{7} + 19806 x^{6} + 16782 x^{5} + 14900 x^{4} + 15322 x^{3} + 15226 x^{2} + 10270 x + 3281 \)

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:  \(58102542838005760000000000=2^{28}\cdot 5^{10}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.76$
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:  $2, 5, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{20} a^{14} - \frac{1}{10} a^{13} - \frac{1}{20} a^{12} + \frac{1}{10} a^{11} - \frac{3}{20} a^{10} + \frac{1}{5} a^{9} + \frac{1}{20} a^{8} + \frac{1}{10} a^{7} - \frac{3}{20} a^{6} - \frac{1}{5} a^{5} + \frac{3}{20} a^{4} + \frac{2}{5} a^{3} - \frac{7}{20} a^{2} - \frac{1}{2} a + \frac{9}{20}$, $\frac{1}{51429780298728103742127380} a^{15} + \frac{82378191597160695626087}{10285956059745620748425476} a^{14} - \frac{44457520091851395325442}{2571489014936405187106369} a^{13} + \frac{524891802923790296411285}{5142978029872810374212738} a^{12} + \frac{6255708263397456782943011}{51429780298728103742127380} a^{11} - \frac{9533526581052508107281957}{51429780298728103742127380} a^{10} + \frac{2463716943651484630899091}{12857445074682025935531845} a^{9} - \frac{2322132839364629845624974}{12857445074682025935531845} a^{8} - \frac{568839901383683892693279}{51429780298728103742127380} a^{7} + \frac{2339489478115833547084859}{10285956059745620748425476} a^{6} - \frac{2061180222717639583401393}{5142978029872810374212738} a^{5} + \frac{5789633666934288901455516}{12857445074682025935531845} a^{4} + \frac{2755349284723426230949619}{51429780298728103742127380} a^{3} + \frac{15236501779843104391962751}{51429780298728103742127380} a^{2} + \frac{69786255072398764201147}{254602872765980711594690} a - \frac{10065768642676772541434201}{25714890149364051871063690}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( -\frac{17454961387319}{17218562792112020} a^{15} + \frac{20014642780936}{4304640698028005} a^{14} - \frac{472283736260913}{17218562792112020} a^{13} + \frac{215958536918623}{8609281396056010} a^{12} - \frac{2495603645972221}{17218562792112020} a^{11} - \frac{1047252735959221}{4304640698028005} a^{10} - \frac{2017106195104191}{3443712558422404} a^{9} - \frac{12330718566612808}{4304640698028005} a^{8} - \frac{60223225952426161}{17218562792112020} a^{7} - \frac{6263797074409033}{4304640698028005} a^{6} - \frac{25441582632006231}{17218562792112020} a^{5} - \frac{14537013575697177}{8609281396056010} a^{4} - \frac{32051486778324699}{17218562792112020} a^{3} - \frac{11628752537031388}{4304640698028005} a^{2} - \frac{14259051392866901}{17218562792112020} a + \frac{4869939415219506}{4304640698028005} \) (order $4$)
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:  \( 1686708.55959 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1048:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 768
The 31 conjugacy class representatives for t16n1048
Character table for t16n1048 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.4.21200.1, 8.0.7191040000.3

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.6.5.2$x^{6} + 10$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
53Data not computed