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Properties

Label 16.4.75610373616...437.14
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![169, 1690, -1443, -7475, -1681, 15527, 16722, 3656, -3197, -1054, -414, 90, 46, -38, 9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 9*x^14 - 38*x^13 + 46*x^12 + 90*x^11 - 414*x^10 - 1054*x^9 - 3197*x^8 + 3656*x^7 + 16722*x^6 + 15527*x^5 - 1681*x^4 - 7475*x^3 - 1443*x^2 + 1690*x + 169)
 
gp: K = bnfinit(x^16 - x^15 + 9*x^14 - 38*x^13 + 46*x^12 + 90*x^11 - 414*x^10 - 1054*x^9 - 3197*x^8 + 3656*x^7 + 16722*x^6 + 15527*x^5 - 1681*x^4 - 7475*x^3 - 1443*x^2 + 1690*x + 169, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 9 x^{14} - 38 x^{13} + 46 x^{12} + 90 x^{11} - 414 x^{10} - 1054 x^{9} - 3197 x^{8} + 3656 x^{7} + 16722 x^{6} + 15527 x^{5} - 1681 x^{4} - 7475 x^{3} - 1443 x^{2} + 1690 x + 169 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13} a^{10} - \frac{5}{13} a^{9} + \frac{2}{13} a^{8} + \frac{4}{13} a^{7} - \frac{1}{13} a^{6} + \frac{6}{13} a^{5} + \frac{5}{13} a^{4} - \frac{3}{13} a^{3} + \frac{4}{13} a^{2}$, $\frac{1}{13} a^{11} + \frac{3}{13} a^{9} + \frac{1}{13} a^{8} + \frac{6}{13} a^{7} + \frac{1}{13} a^{6} - \frac{4}{13} a^{5} - \frac{4}{13} a^{4} + \frac{2}{13} a^{3} - \frac{6}{13} a^{2}$, $\frac{1}{13} a^{12} + \frac{3}{13} a^{9} + \frac{2}{13} a^{7} - \frac{1}{13} a^{6} + \frac{4}{13} a^{5} + \frac{3}{13} a^{3} + \frac{1}{13} a^{2}$, $\frac{1}{13} a^{13} + \frac{2}{13} a^{9} - \frac{4}{13} a^{8} - \frac{6}{13} a^{6} - \frac{5}{13} a^{5} + \frac{1}{13} a^{4} - \frac{3}{13} a^{3} + \frac{1}{13} a^{2}$, $\frac{1}{1183} a^{14} + \frac{38}{1183} a^{13} - \frac{4}{1183} a^{12} - \frac{38}{1183} a^{11} - \frac{6}{1183} a^{10} - \frac{300}{1183} a^{9} - \frac{323}{1183} a^{8} - \frac{40}{1183} a^{7} + \frac{28}{169} a^{6} + \frac{29}{1183} a^{5} - \frac{256}{1183} a^{4} - \frac{229}{1183} a^{3} + \frac{282}{1183} a^{2} - \frac{8}{91} a - \frac{23}{91}$, $\frac{1}{1727922543246610682074103} a^{15} + \frac{352563605416538112801}{1727922543246610682074103} a^{14} + \frac{61852385723757778348679}{1727922543246610682074103} a^{13} - \frac{425528859440835855344}{12083374428297976797721} a^{12} + \frac{51095270960024813122}{35263725372379809838247} a^{11} - \frac{26134361068578923384428}{1727922543246610682074103} a^{10} + \frac{128094106762382617868666}{1727922543246610682074103} a^{9} - \frac{59632193647603656599393}{157083867567873698370373} a^{8} - \frac{518802028997988415787818}{1727922543246610682074103} a^{7} - \frac{20451424612498837595996}{132917118711277744774931} a^{6} + \frac{155449238163492565865854}{1727922543246610682074103} a^{5} - \frac{535851443750945646666206}{1727922543246610682074103} a^{4} + \frac{169284104418719728212717}{1727922543246610682074103} a^{3} - \frac{81478226044619041781027}{1727922543246610682074103} a^{2} - \frac{2561577870297134584725}{18988159815896820682133} a - \frac{49134000798720333493266}{132917118711277744774931}$

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:  \( 66214254.4391 \) (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, 8.4.11944081475573.1

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 $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
53Data not computed