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Properties

Label 16.0.47763940103...625.18
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 89^{14}$
Root discriminant $169.80$
Ramified primes $5, 89$
Class number $16384$ (GRH)
Class group $[4, 8, 8, 8, 8]$ (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![1051559375, -453825000, 371963750, 4094000, 42649275, 11861450, 6028400, 2483375, 706440, 138170, -7005, -4890, -254, -267, 64, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 64*x^14 - 267*x^13 - 254*x^12 - 4890*x^11 - 7005*x^10 + 138170*x^9 + 706440*x^8 + 2483375*x^7 + 6028400*x^6 + 11861450*x^5 + 42649275*x^4 + 4094000*x^3 + 371963750*x^2 - 453825000*x + 1051559375)
 
gp: K = bnfinit(x^16 - 3*x^15 + 64*x^14 - 267*x^13 - 254*x^12 - 4890*x^11 - 7005*x^10 + 138170*x^9 + 706440*x^8 + 2483375*x^7 + 6028400*x^6 + 11861450*x^5 + 42649275*x^4 + 4094000*x^3 + 371963750*x^2 - 453825000*x + 1051559375, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 64 x^{14} - 267 x^{13} - 254 x^{12} - 4890 x^{11} - 7005 x^{10} + 138170 x^{9} + 706440 x^{8} + 2483375 x^{7} + 6028400 x^{6} + 11861450 x^{5} + 42649275 x^{4} + 4094000 x^{3} + 371963750 x^{2} - 453825000 x + 1051559375 \)

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:  \(477639401035264016929733847900390625=5^{12}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $169.80$
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:  $5, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4}$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{4}$, $\frac{1}{25} a^{10} - \frac{1}{25} a^{9} + \frac{2}{25} a^{8} + \frac{2}{25} a^{7} - \frac{2}{5} a^{3}$, $\frac{1}{25} a^{11} + \frac{1}{25} a^{9} - \frac{1}{25} a^{8} + \frac{2}{25} a^{7} + \frac{2}{5} a^{5} - \frac{2}{5} a^{3}$, $\frac{1}{150} a^{12} - \frac{1}{30} a^{9} + \frac{4}{75} a^{7} - \frac{1}{30} a^{6} - \frac{1}{15} a^{5} - \frac{1}{3} a^{4} + \frac{7}{30} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{150} a^{13} + \frac{1}{150} a^{10} - \frac{1}{25} a^{9} - \frac{1}{15} a^{8} + \frac{7}{150} a^{7} - \frac{1}{15} a^{6} + \frac{1}{15} a^{5} - \frac{11}{30} a^{4} - \frac{1}{15} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a$, $\frac{1}{750} a^{14} + \frac{1}{375} a^{13} - \frac{1}{750} a^{12} + \frac{13}{750} a^{11} - \frac{2}{375} a^{10} - \frac{7}{150} a^{9} + \frac{1}{150} a^{8} - \frac{4}{75} a^{7} - \frac{7}{150} a^{6} - \frac{1}{30} a^{5} + \frac{7}{15} a^{4} - \frac{1}{6} a^{3} + \frac{1}{30} a^{2} - \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{10766572817085387410855935059279159950181930889225991250} a^{15} + \frac{2866178722578421666777210607878085521809556519689121}{5383286408542693705427967529639579975090965444612995625} a^{14} + \frac{11514275696630379941342338683215797321042006761027727}{5383286408542693705427967529639579975090965444612995625} a^{13} + \frac{31608850200074537169063806280990173712116873500053013}{10766572817085387410855935059279159950181930889225991250} a^{12} + \frac{52482408440291115363338490921985327629173707992845128}{5383286408542693705427967529639579975090965444612995625} a^{11} - \frac{10477034959700278653623586700826494236278420488786877}{1076657281708538741085593505927915995018193088922599125} a^{10} + \frac{66326867484354075059984541191972412850494910733647429}{2153314563417077482171187011855831990036386177845198250} a^{9} + \frac{15281917391338861417927941176602724695103574858442397}{1076657281708538741085593505927915995018193088922599125} a^{8} - \frac{54596212526585242257355064311529421836770101652289081}{1076657281708538741085593505927915995018193088922599125} a^{7} - \frac{32765800174729181994584988215595850428983324884850773}{430662912683415496434237402371166398007277235569039650} a^{6} - \frac{9269836885341579397690444050598024930886350759816352}{215331456341707748217118701185583199003638617784519825} a^{5} + \frac{66172274038224802863702758718449319621160472670540539}{215331456341707748217118701185583199003638617784519825} a^{4} - \frac{62116247870787725086625308639634814193839861380223689}{430662912683415496434237402371166398007277235569039650} a^{3} - \frac{11794999622010101480391416804409982672036131552613582}{43066291268341549643423740237116639800727723556903965} a^{2} + \frac{3167888248970606067500192540390041414249935969962441}{8613258253668309928684748047423327960145544711380793} a + \frac{514696442179737982855958453155904479241381350134851}{2871086084556103309561582682474442653381848237126931}$

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

Class group and class number

$C_{4}\times C_{8}\times C_{8}\times C_{8}\times C_{8}$, which has order $16384$ (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:  \( 14693160.0683 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{445}) \), \(\Q(\sqrt{5}, \sqrt{89})\), 4.4.3524845.1 x2, 4.4.17624225.1 x2, 8.8.310613306850625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$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}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$89$89.8.7.2$x^{8} - 801$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.2$x^{8} - 801$$8$$1$$7$$C_8$$[\ ]_{8}$