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Properties

Label 16.0.48789140945...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{10}\cdot 41^{6}$
Root discriminant $110.41$
Ramified primes $5, 29, 41$
Class number $144$ (GRH)
Class group $[2, 2, 36]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T516)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![40301591, -89199484, 182698492, -154398062, 85920234, -25497960, 4001937, 50279, 50527, -74414, 30285, -3593, 120, 76, 21, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 21*x^14 + 76*x^13 + 120*x^12 - 3593*x^11 + 30285*x^10 - 74414*x^9 + 50527*x^8 + 50279*x^7 + 4001937*x^6 - 25497960*x^5 + 85920234*x^4 - 154398062*x^3 + 182698492*x^2 - 89199484*x + 40301591)
 
gp: K = bnfinit(x^16 - 5*x^15 + 21*x^14 + 76*x^13 + 120*x^12 - 3593*x^11 + 30285*x^10 - 74414*x^9 + 50527*x^8 + 50279*x^7 + 4001937*x^6 - 25497960*x^5 + 85920234*x^4 - 154398062*x^3 + 182698492*x^2 - 89199484*x + 40301591, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 21 x^{14} + 76 x^{13} + 120 x^{12} - 3593 x^{11} + 30285 x^{10} - 74414 x^{9} + 50527 x^{8} + 50279 x^{7} + 4001937 x^{6} - 25497960 x^{5} + 85920234 x^{4} - 154398062 x^{3} + 182698492 x^{2} - 89199484 x + 40301591 \)

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:  \(487891409452798143616318603515625=5^{12}\cdot 29^{10}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $110.41$
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:  $5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{44} a^{13} + \frac{5}{44} a^{12} - \frac{1}{11} a^{11} - \frac{9}{44} a^{10} + \frac{1}{4} a^{9} + \frac{19}{44} a^{8} + \frac{4}{11} a^{7} + \frac{3}{11} a^{6} + \frac{1}{22} a^{5} + \frac{5}{11} a^{4} + \frac{1}{4} a^{3} + \frac{7}{22} a^{2} - \frac{21}{44} a$, $\frac{1}{176} a^{14} - \frac{9}{88} a^{12} - \frac{1}{8} a^{11} - \frac{2}{11} a^{10} - \frac{47}{176} a^{9} + \frac{21}{88} a^{8} + \frac{53}{176} a^{7} + \frac{37}{88} a^{6} + \frac{27}{88} a^{5} - \frac{23}{176} a^{4} - \frac{85}{176} a^{3} - \frac{7}{88} a^{2} + \frac{39}{176} a + \frac{3}{16}$, $\frac{1}{6002378348384833273941496397650470902955971532472825936} a^{15} + \frac{307200007319452632398939012375129444724527955177631}{157957324957495612472144642043433444814630829801916472} a^{14} + \frac{17986978897940778745259631083745759165209917709867959}{3001189174192416636970748198825235451477985766236412968} a^{13} - \frac{327446433726973112053739263374608329400566403152492755}{3001189174192416636970748198825235451477985766236412968} a^{12} + \frac{21634230742086474574071996082518609782888218190767261}{750297293548104159242687049706308862869496441559103242} a^{11} + \frac{1387425008749579901629744435423233135664205443543638977}{6002378348384833273941496397650470902955971532472825936} a^{10} + \frac{1470334856083298819586514664709902754540299401926007}{3714343037366852273478648760922321103314338819599521} a^{9} - \frac{133601539212675668006250251636668791928239962451906009}{315914649914991224944289284086866889629261659603832944} a^{8} - \frac{371722558222242784896626059993387632421393883091827813}{750297293548104159242687049706308862869496441559103242} a^{7} + \frac{75182433211902081512962996316006025245898505306882669}{3001189174192416636970748198825235451477985766236412968} a^{6} + \frac{28688304734230467022215233244455014294232864450380449}{59429488597869636375658380174757137653029421113592336} a^{5} + \frac{2516097868717240031902662130629262667722970042985880189}{6002378348384833273941496397650470902955971532472825936} a^{4} - \frac{2366406812503158166915632821247339437764484437714210}{375148646774052079621343524853154431434748220779551621} a^{3} + \frac{2516867238749741605797903277484219280582718951510630087}{6002378348384833273941496397650470902955971532472825936} a^{2} - \frac{692595763863002385586672953908174918276065364111125633}{6002378348384833273941496397650470902955971532472825936} a + \frac{52755328288406945013334253645403570379528904211837393}{272835379472037876088249836256839586497998706021492088}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{36}$, which has order $144$ (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:  \( 294954156.828 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T516):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.862025.1, 4.0.3625.1, 4.4.148625.2, 8.0.18577177515625.8

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed