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Properties

Label 16.4.75610373616...437.10
Degree $16$
Signature $[4, 6]$
Discriminant $13^{8}\cdot 53^{11}$
Root discriminant $55.26$
Ramified primes $13, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4:D_4.D_4$ (as 16T681)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-211, 3144, -6187, 2635, 8549, -12344, 11329, -8466, 5157, -3363, 1709, -879, 337, -117, 31, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 31*x^14 - 117*x^13 + 337*x^12 - 879*x^11 + 1709*x^10 - 3363*x^9 + 5157*x^8 - 8466*x^7 + 11329*x^6 - 12344*x^5 + 8549*x^4 + 2635*x^3 - 6187*x^2 + 3144*x - 211)
 
gp: K = bnfinit(x^16 - 6*x^15 + 31*x^14 - 117*x^13 + 337*x^12 - 879*x^11 + 1709*x^10 - 3363*x^9 + 5157*x^8 - 8466*x^7 + 11329*x^6 - 12344*x^5 + 8549*x^4 + 2635*x^3 - 6187*x^2 + 3144*x - 211, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 31 x^{14} - 117 x^{13} + 337 x^{12} - 879 x^{11} + 1709 x^{10} - 3363 x^{9} + 5157 x^{8} - 8466 x^{7} + 11329 x^{6} - 12344 x^{5} + 8549 x^{4} + 2635 x^{3} - 6187 x^{2} + 3144 x - 211 \)

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:  \(7561037361641682928770951437=13^{8}\cdot 53^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.26$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{6}{13} a^{13} + \frac{5}{13} a^{12} + \frac{5}{13} a^{11} - \frac{2}{13} a^{10} - \frac{2}{13} a^{9} - \frac{4}{13} a^{8} - \frac{4}{13} a^{7} + \frac{2}{13} a^{6} - \frac{3}{13} a^{5} - \frac{5}{13} a^{4} + \frac{6}{13} a^{3} - \frac{2}{13} a^{2} - \frac{5}{13} a - \frac{6}{13}$, $\frac{1}{395833162335188206963394489} a^{15} + \frac{6031003515207839319712}{35984832939562564269399499} a^{14} + \frac{125142917803361388255436329}{395833162335188206963394489} a^{13} + \frac{74039844347588394159724392}{395833162335188206963394489} a^{12} - \frac{8591225872043426343098580}{30448704795014477458722653} a^{11} + \frac{157708987099750833130417987}{395833162335188206963394489} a^{10} + \frac{70108406792647187374740008}{395833162335188206963394489} a^{9} - \frac{85410766248024932078592309}{395833162335188206963394489} a^{8} + \frac{156380567299911114412537886}{395833162335188206963394489} a^{7} + \frac{44526597861989609476982654}{395833162335188206963394489} a^{6} - \frac{9876826192555426242089841}{35984832939562564269399499} a^{5} + \frac{52571535057568232844616300}{395833162335188206963394489} a^{4} + \frac{63011156707446203566182863}{395833162335188206963394489} a^{3} - \frac{189764648448127250832179064}{395833162335188206963394489} a^{2} + \frac{196565897522133556288168133}{395833162335188206963394489} a + \frac{67062319969181471534934030}{395833162335188206963394489}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 48447009.819 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:D_4.D_4$ (as 16T681):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 19 conjugacy class representatives for $C_4:D_4.D_4$
Character table for $C_4:D_4.D_4$

Intermediate fields

\(\Q(\sqrt{53}) \), 4.4.36517.1, 8.4.70675038317.1

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 $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
53Data not computed