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Properties

Label 16.0.12251765391...000.56
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 5^{8}\cdot 19^{8}$
Root discriminant $135.06$
Ramified primes $2, 3, 5, 19$
Class number $3088384$ (GRH)
Class group $[2, 16, 16, 6032]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![75639875521, -27285235712, 31536986508, -8904466088, 5479602212, -1303648472, 555347828, -109181072, 34483221, -5008984, 1221920, -124040, 24010, -1568, 244, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 244*x^14 - 1568*x^13 + 24010*x^12 - 124040*x^11 + 1221920*x^10 - 5008984*x^9 + 34483221*x^8 - 109181072*x^7 + 555347828*x^6 - 1303648472*x^5 + 5479602212*x^4 - 8904466088*x^3 + 31536986508*x^2 - 27285235712*x + 75639875521)
 
gp: K = bnfinit(x^16 - 8*x^15 + 244*x^14 - 1568*x^13 + 24010*x^12 - 124040*x^11 + 1221920*x^10 - 5008984*x^9 + 34483221*x^8 - 109181072*x^7 + 555347828*x^6 - 1303648472*x^5 + 5479602212*x^4 - 8904466088*x^3 + 31536986508*x^2 - 27285235712*x + 75639875521, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 244 x^{14} - 1568 x^{13} + 24010 x^{12} - 124040 x^{11} + 1221920 x^{10} - 5008984 x^{9} + 34483221 x^{8} - 109181072 x^{7} + 555347828 x^{6} - 1303648472 x^{5} + 5479602212 x^{4} - 8904466088 x^{3} + 31536986508 x^{2} - 27285235712 x + 75639875521 \)

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:  \(12251765391792386665453977600000000=2^{48}\cdot 3^{8}\cdot 5^{8}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $135.06$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4560=2^{4}\cdot 3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4560}(1,·)$, $\chi_{4560}(2051,·)$, $\chi_{4560}(3649,·)$, $\chi_{4560}(2509,·)$, $\chi_{4560}(911,·)$, $\chi_{4560}(4559,·)$, $\chi_{4560}(1369,·)$, $\chi_{4560}(3419,·)$, $\chi_{4560}(3421,·)$, $\chi_{4560}(229,·)$, $\chi_{4560}(2279,·)$, $\chi_{4560}(2281,·)$, $\chi_{4560}(4331,·)$, $\chi_{4560}(1139,·)$, $\chi_{4560}(1141,·)$, $\chi_{4560}(3191,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{29} a^{8} - \frac{4}{29} a^{7} - \frac{2}{29} a^{6} - \frac{9}{29} a^{5} - \frac{10}{29} a^{4} + \frac{11}{29} a^{3} - \frac{14}{29} a^{2} - \frac{2}{29} a + \frac{7}{29}$, $\frac{1}{29} a^{9} + \frac{11}{29} a^{7} + \frac{12}{29} a^{6} + \frac{12}{29} a^{5} + \frac{1}{29} a^{3} - \frac{1}{29} a - \frac{1}{29}$, $\frac{1}{13021} a^{10} - \frac{5}{13021} a^{9} + \frac{173}{13021} a^{8} - \frac{662}{13021} a^{7} + \frac{6443}{13021} a^{6} - \frac{4012}{13021} a^{5} + \frac{3456}{13021} a^{4} - \frac{5328}{13021} a^{3} - \frac{2240}{13021} a^{2} + \frac{2174}{13021} a - \frac{2486}{13021}$, $\frac{1}{13021} a^{11} + \frac{148}{13021} a^{9} + \frac{7}{449} a^{8} + \frac{3133}{13021} a^{7} + \frac{2161}{13021} a^{6} - \frac{3583}{13021} a^{5} - \frac{1069}{13021} a^{4} - \frac{2838}{13021} a^{3} + \frac{3995}{13021} a^{2} - \frac{4637}{13021} a + \frac{591}{13021}$, $\frac{1}{299483} a^{12} - \frac{6}{299483} a^{11} - \frac{1}{299483} a^{10} + \frac{60}{299483} a^{9} + \frac{1731}{299483} a^{8} - \frac{7350}{299483} a^{7} - \frac{12104}{299483} a^{6} + \frac{62355}{299483} a^{5} + \frac{53025}{299483} a^{4} + \frac{80780}{299483} a^{3} + \frac{11956}{299483} a^{2} + \frac{109036}{299483} a + \frac{142368}{299483}$, $\frac{1}{299483} a^{13} + \frac{9}{299483} a^{11} + \frac{8}{299483} a^{10} - \frac{1198}{299483} a^{9} + \frac{192}{13021} a^{8} + \frac{4769}{299483} a^{7} - \frac{31682}{299483} a^{6} + \frac{23482}{299483} a^{5} - \frac{108703}{299483} a^{4} + \frac{1883}{299483} a^{3} - \frac{131384}{299483} a^{2} - \frac{105361}{299483} a + \frac{107628}{299483}$, $\frac{1}{845017130875590609923} a^{14} - \frac{1}{120716732982227229989} a^{13} - \frac{604550105925760}{845017130875590609923} a^{12} + \frac{3627300635554651}{845017130875590609923} a^{11} - \frac{457955010534526}{29138521754330710687} a^{10} + \frac{33153220701588469}{845017130875590609923} a^{9} - \frac{7703519024899150755}{845017130875590609923} a^{8} + \frac{30655057082378173653}{845017130875590609923} a^{7} + \frac{351949828761666900689}{845017130875590609923} a^{6} - \frac{318025711977778129821}{845017130875590609923} a^{5} - \frac{155413844764406541249}{845017130875590609923} a^{4} - \frac{250101088034143937975}{845017130875590609923} a^{3} - \frac{8390302665512254413}{36739875255460461301} a^{2} - \frac{303423786887551789066}{845017130875590609923} a - \frac{37336403704453009279}{120716732982227229989}$, $\frac{1}{56855930674738959331903360403} a^{15} + \frac{33641873}{56855930674738959331903360403} a^{14} - \frac{37459606094260311549400}{56855930674738959331903360403} a^{13} + \frac{164050732609730198910}{280078476230241178974893401} a^{12} + \frac{181111298688135030612962}{8122275810676994190271908629} a^{11} + \frac{8661501696665877973452}{353142426551173660446604723} a^{10} + \frac{132657196472960272719003434}{8122275810676994190271908629} a^{9} + \frac{770181569053305691903945610}{56855930674738959331903360403} a^{8} + \frac{593884925314405764119380611}{56855930674738959331903360403} a^{7} - \frac{97616932131685388123695106}{280078476230241178974893401} a^{6} + \frac{91313989369162379773013142}{8122275810676994190271908629} a^{5} + \frac{229133579296812334829810809}{8122275810676994190271908629} a^{4} + \frac{1345437877109926181007865822}{8122275810676994190271908629} a^{3} + \frac{789401306995059809225692358}{8122275810676994190271908629} a^{2} + \frac{22563607640199794406259360338}{56855930674738959331903360403} a - \frac{442710137982858654522447}{1071964604814173708627677}$

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

Class group and class number

$C_{2}\times C_{16}\times C_{16}\times C_{6032}$, which has order $3088384$ (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:  \( 12198.951274811623 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-285}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-114}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-570}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-57}) \), \(\Q(\sqrt{10}, \sqrt{-114})\), \(\Q(\sqrt{2}, \sqrt{-285})\), \(\Q(\sqrt{5}, \sqrt{-57})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{10}, \sqrt{-57})\), \(\Q(\sqrt{2}, \sqrt{-57})\), \(\Q(\sqrt{5}, \sqrt{-114})\), 4.0.166348800.2, \(\Q(\zeta_{16})^+\), 4.0.6653952.2, 4.4.51200.1, 8.0.432373800960000.208, 8.0.110687693045760000.20, 8.0.110687693045760000.6, 8.0.27671923261440000.256, 8.8.2621440000.1, 8.0.110687693045760000.11, 8.0.177100308873216.55

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 R R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$