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Properties

Label 16.0.67233141694...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{14}\cdot 29^{6}\cdot 41^{4}$
Root discriminant $73.15$
Ramified primes $2, 5, 29, 41$
Class number $32$ (GRH)
Class group $[2, 2, 2, 4]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T646)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1903906301, 378955036, -1002864095, -156467770, 239383365, 22108018, -29643917, -1034380, 1885080, 31830, -73287, -2902, 2815, -100, -10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 10*x^14 - 100*x^13 + 2815*x^12 - 2902*x^11 - 73287*x^10 + 31830*x^9 + 1885080*x^8 - 1034380*x^7 - 29643917*x^6 + 22108018*x^5 + 239383365*x^4 - 156467770*x^3 - 1002864095*x^2 + 378955036*x + 1903906301)
 
gp: K = bnfinit(x^16 - 4*x^15 - 10*x^14 - 100*x^13 + 2815*x^12 - 2902*x^11 - 73287*x^10 + 31830*x^9 + 1885080*x^8 - 1034380*x^7 - 29643917*x^6 + 22108018*x^5 + 239383365*x^4 - 156467770*x^3 - 1002864095*x^2 + 378955036*x + 1903906301, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 10 x^{14} - 100 x^{13} + 2815 x^{12} - 2902 x^{11} - 73287 x^{10} + 31830 x^{9} + 1885080 x^{8} - 1034380 x^{7} - 29643917 x^{6} + 22108018 x^{5} + 239383365 x^{4} - 156467770 x^{3} - 1002864095 x^{2} + 378955036 x + 1903906301 \)

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:  \(672331416948912400000000000000=2^{16}\cdot 5^{14}\cdot 29^{6}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.15$
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:  $2, 5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{5} + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{10} - \frac{1}{10} a^{8} + \frac{2}{5} a^{7} - \frac{3}{10} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{3}{10}$, $\frac{1}{410} a^{13} - \frac{3}{410} a^{12} - \frac{13}{410} a^{11} - \frac{19}{410} a^{10} + \frac{13}{410} a^{9} - \frac{11}{410} a^{8} - \frac{13}{410} a^{7} - \frac{89}{410} a^{6} + \frac{13}{41} a^{5} + \frac{94}{205} a^{4} + \frac{24}{205} a^{3} + \frac{13}{41} a^{2} + \frac{199}{410} a - \frac{171}{410}$, $\frac{1}{13530} a^{14} - \frac{1}{2706} a^{13} + \frac{97}{2706} a^{12} + \frac{991}{13530} a^{11} - \frac{1261}{13530} a^{10} - \frac{1349}{13530} a^{9} + \frac{829}{13530} a^{8} + \frac{471}{4510} a^{7} - \frac{2224}{6765} a^{6} + \frac{247}{1353} a^{5} - \frac{1}{33} a^{4} + \frac{2477}{6765} a^{3} + \frac{1401}{4510} a^{2} + \frac{357}{4510} a - \frac{182}{615}$, $\frac{1}{2081531093679178465063727082937074657999422657079305547930} a^{15} - \frac{883901240183790519465545542786805700211151749199813}{37846019885075972092067765144310448327262230128714646326} a^{14} - \frac{2394558820082447515721833286210168393305059461097104299}{2081531093679178465063727082937074657999422657079305547930} a^{13} + \frac{8025008326844596655956069106453515869781620287770080578}{1040765546839589232531863541468537328999711328539652773965} a^{12} + \frac{18405239704001384210095790925151453220619106751853139367}{2081531093679178465063727082937074657999422657079305547930} a^{11} + \frac{93655513537411832091661272946376996056771691218603854362}{1040765546839589232531863541468537328999711328539652773965} a^{10} + \frac{5327516119332166165534801424525968088188245516137114467}{2081531093679178465063727082937074657999422657079305547930} a^{9} + \frac{2184905316342915241432830031403562347208437560423026150}{69384369789305948835457569431235821933314088569310184931} a^{8} - \frac{312850079109092535966003858419760950610392973242205714223}{1040765546839589232531863541468537328999711328539652773965} a^{7} + \frac{26142993583224019503465765271840631133797886236935655317}{2081531093679178465063727082937074657999422657079305547930} a^{6} - \frac{16425385379450246763466883122068798996469915558250291329}{1040765546839589232531863541468537328999711328539652773965} a^{5} + \frac{46064730961093432071213293321127685108493313410203676641}{94615049712689930230169412860776120818155575321786615815} a^{4} - \frac{108652992196745688334769614559630114449116756669273024837}{693843697893059488354575694312358219333140885693101849310} a^{3} + \frac{54419375523717468202074204877178905864595291787369914771}{693843697893059488354575694312358219333140885693101849310} a^{2} - \frac{63827255742045369716569574915781665128231689495526821953}{208153109367917846506372708293707465799942265707930554793} a - \frac{135205213997106703630055919991986446965572886455842167}{417726488797748036336288798502322829219229913120470710}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $32$ (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:  \( -\frac{68992932009908732621419095825480738147102942061}{4615368278667801474642410383452493698446613430331054430} a^{15} + \frac{195583880281588859893173859683054043894801551607}{2307684139333900737321205191726246849223306715165527215} a^{14} - \frac{83142692578451816329144139917602211424374177072}{2307684139333900737321205191726246849223306715165527215} a^{13} + \frac{4198543154770029454827872580109084158149939017783}{2307684139333900737321205191726246849223306715165527215} a^{12} - \frac{104073457361792550194971156492948192492382417006719}{2307684139333900737321205191726246849223306715165527215} a^{11} + \frac{286220073710455501699364817431725232404185005353948}{2307684139333900737321205191726246849223306715165527215} a^{10} + \frac{1735724063261069833308869689224548010127919690428369}{2307684139333900737321205191726246849223306715165527215} a^{9} - \frac{1040906649304712037442361068667642061479634794828968}{769228046444633579107068397242082283074435571721842405} a^{8} - \frac{107820072161794950520764624525200656955445281801033413}{4615368278667801474642410383452493698446613430331054430} a^{7} + \frac{117272825580979686623806698384465952336119748622489452}{2307684139333900737321205191726246849223306715165527215} a^{6} + \frac{129923436422289963814399105843356835750644771371089817}{461536827866780147464241038345249369844661343033105443} a^{5} - \frac{1528669138973975171358515691407652490923554681312053643}{2307684139333900737321205191726246849223306715165527215} a^{4} - \frac{2256036451570707305029343185609338402241617136745854713}{1538456092889267158214136794484164566148871143443684810} a^{3} + \frac{2483786552020206044857834084146716931954506200295339946}{769228046444633579107068397242082283074435571721842405} a^{2} + \frac{15961189670399514639775947494347545209563881927030335237}{4615368278667801474642410383452493698446613430331054430} a - \frac{32594502378059446644115934741572696176826407396969132}{5094225473143268735808399981735644258771096501469155} \) (order $10$)
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:  \( 32219502.8941 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T646):

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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.58000.1, 4.0.11600.1, 8.0.3364000000.5

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
5Data not computed
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed