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Properties

Label 16.0.20273287048...2041.6
Degree $16$
Signature $[0, 8]$
Discriminant $23^{12}\cdot 41^{13}$
Root discriminant $214.63$
Ramified primes $23, 41$
Class number $384$ (GRH)
Class group $[2, 2, 2, 4, 12]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64667028003, -144221692018, 118568037753, -39285105450, 1551213890, 1424825846, 48261497, -97134971, 11589258, -425048, 160951, -25713, 2357, -247, -26, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 26*x^14 - 247*x^13 + 2357*x^12 - 25713*x^11 + 160951*x^10 - 425048*x^9 + 11589258*x^8 - 97134971*x^7 + 48261497*x^6 + 1424825846*x^5 + 1551213890*x^4 - 39285105450*x^3 + 118568037753*x^2 - 144221692018*x + 64667028003)
 
gp: K = bnfinit(x^16 - 2*x^15 - 26*x^14 - 247*x^13 + 2357*x^12 - 25713*x^11 + 160951*x^10 - 425048*x^9 + 11589258*x^8 - 97134971*x^7 + 48261497*x^6 + 1424825846*x^5 + 1551213890*x^4 - 39285105450*x^3 + 118568037753*x^2 - 144221692018*x + 64667028003, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 26 x^{14} - 247 x^{13} + 2357 x^{12} - 25713 x^{11} + 160951 x^{10} - 425048 x^{9} + 11589258 x^{8} - 97134971 x^{7} + 48261497 x^{6} + 1424825846 x^{5} + 1551213890 x^{4} - 39285105450 x^{3} + 118568037753 x^{2} - 144221692018 x + 64667028003 \)

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:  \(20273287048130392461673121562633432041=23^{12}\cdot 41^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $214.63$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{41} a^{12} + \frac{14}{41} a^{11} + \frac{20}{41} a^{10} + \frac{10}{41} a^{9} - \frac{11}{41} a^{8} - \frac{16}{41} a^{7} - \frac{12}{41} a^{6} - \frac{15}{41} a^{5} - \frac{6}{41} a^{4} + \frac{5}{41} a^{3} - \frac{2}{41} a^{2} - \frac{16}{41} a + \frac{10}{41}$, $\frac{1}{123} a^{13} - \frac{1}{123} a^{12} - \frac{26}{123} a^{11} - \frac{44}{123} a^{10} + \frac{44}{123} a^{9} + \frac{26}{123} a^{8} - \frac{59}{123} a^{7} - \frac{40}{123} a^{6} + \frac{55}{123} a^{5} + \frac{18}{41} a^{4} + \frac{46}{123} a^{3} + \frac{14}{123} a^{2} - \frac{37}{123} a - \frac{9}{41}$, $\frac{1}{2829} a^{14} - \frac{9}{943} a^{12} - \frac{439}{2829} a^{11} + \frac{10}{23} a^{10} + \frac{439}{2829} a^{9} - \frac{11}{943} a^{8} - \frac{33}{943} a^{7} - \frac{118}{943} a^{6} - \frac{1244}{2829} a^{5} - \frac{1376}{2829} a^{4} - \frac{21}{943} a^{3} - \frac{1253}{2829} a^{2} - \frac{1294}{2829} a + \frac{32}{943}$, $\frac{1}{26819051652315355987890196716414988565729647479916869287728635775697723121} a^{15} + \frac{459622877843847506602901586070426692672283399522949745581988532312208}{26819051652315355987890196716414988565729647479916869287728635775697723121} a^{14} + \frac{2019824372017133988798285799891694022863830827636374798898808789407559}{2979894628035039554210021857379443173969960831101874365303181752855302569} a^{13} + \frac{306577653214690579519359578402508675168260401277653021175580683513242745}{26819051652315355987890196716414988565729647479916869287728635775697723121} a^{12} + \frac{3687269727992438976341865126478083211365475563444317845459907211417792625}{26819051652315355987890196716414988565729647479916869287728635775697723121} a^{11} + \frac{7708308941743314345115226495513085480866975008769116864520572499651858842}{26819051652315355987890196716414988565729647479916869287728635775697723121} a^{10} + \frac{11723190706322636874079550884100191842920213428096310768619949603216749905}{26819051652315355987890196716414988565729647479916869287728635775697723121} a^{9} - \frac{2843374688912204017255939019568936043879358095165690085785384287647081486}{8939683884105118662630065572138329521909882493305623095909545258565907707} a^{8} + \frac{2568003275926205932804500595731156552479266515032868805040206933714338632}{8939683884105118662630065572138329521909882493305623095909545258565907707} a^{7} - \frac{1307213935902099620987870669922155094835165963120612528395911968710889988}{26819051652315355987890196716414988565729647479916869287728635775697723121} a^{6} + \frac{3273897349592077417879564689868636064755958782327330709084032729306763041}{8939683884105118662630065572138329521909882493305623095909545258565907707} a^{5} - \frac{5484748372753082439241777511501002105276665052003726677687768842448943870}{26819051652315355987890196716414988565729647479916869287728635775697723121} a^{4} - \frac{7929030782899412704117960718415215488511686021936846036855301052561302642}{26819051652315355987890196716414988565729647479916869287728635775697723121} a^{3} + \frac{6136841710717014951066242356445208503348144341458090613538297614822020118}{26819051652315355987890196716414988565729647479916869287728635775697723121} a^{2} + \frac{6932499056370812968426767880404571239902074530591062938895933799469656730}{26819051652315355987890196716414988565729647479916869287728635775697723121} a - \frac{3387120036579100652638804171580210569175480031458615637136836640892012282}{8939683884105118662630065572138329521909882493305623095909545258565907707}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{12}$, which has order $384$ (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:  \( 10710950383.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-23}) \), 4.0.21689.1, 8.0.32421315144041.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ $16$ $16$ R ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$23$23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
41Data not computed