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Properties

Label 16.0.60283401692...0625.5
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 61^{14}$
Root discriminant $149.20$
Ramified primes $5, 61$
Class number $48400$ (GRH)
Class group $[2, 110, 220]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![913931536, 228573380, 174226747, 92075725, 26373403, 5719110, 4876594, 389090, 195560, 111440, -3876, 4690, 518, -30, 37, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 37*x^14 - 30*x^13 + 518*x^12 + 4690*x^11 - 3876*x^10 + 111440*x^9 + 195560*x^8 + 389090*x^7 + 4876594*x^6 + 5719110*x^5 + 26373403*x^4 + 92075725*x^3 + 174226747*x^2 + 228573380*x + 913931536)
 
gp: K = bnfinit(x^16 - 5*x^15 + 37*x^14 - 30*x^13 + 518*x^12 + 4690*x^11 - 3876*x^10 + 111440*x^9 + 195560*x^8 + 389090*x^7 + 4876594*x^6 + 5719110*x^5 + 26373403*x^4 + 92075725*x^3 + 174226747*x^2 + 228573380*x + 913931536, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 37 x^{14} - 30 x^{13} + 518 x^{12} + 4690 x^{11} - 3876 x^{10} + 111440 x^{9} + 195560 x^{8} + 389090 x^{7} + 4876594 x^{6} + 5719110 x^{5} + 26373403 x^{4} + 92075725 x^{3} + 174226747 x^{2} + 228573380 x + 913931536 \)

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:  \(60283401692879138764114019775390625=5^{14}\cdot 61^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $149.20$
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:  $5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{16} a^{8} + \frac{3}{16} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{3}{16} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{3}{16} a^{6} + \frac{1}{8} a^{4} - \frac{3}{16} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{448} a^{13} + \frac{3}{448} a^{12} - \frac{13}{448} a^{11} - \frac{13}{448} a^{10} + \frac{53}{448} a^{9} + \frac{7}{64} a^{8} + \frac{1}{64} a^{7} + \frac{7}{64} a^{6} - \frac{111}{448} a^{5} - \frac{95}{448} a^{4} - \frac{153}{448} a^{3} + \frac{155}{448} a^{2} + \frac{1}{112} a - \frac{1}{4}$, $\frac{1}{20608} a^{14} + \frac{3}{5152} a^{13} - \frac{15}{1472} a^{12} + \frac{117}{10304} a^{11} - \frac{71}{2576} a^{10} - \frac{717}{10304} a^{9} + \frac{75}{736} a^{8} - \frac{1}{92} a^{7} - \frac{1739}{10304} a^{6} - \frac{2017}{10304} a^{5} - \frac{57}{368} a^{4} + \frac{2497}{10304} a^{3} - \frac{5293}{20608} a^{2} - \frac{1447}{5152} a - \frac{13}{184}$, $\frac{1}{9796698496992647504202162047355478813427560300672} a^{15} - \frac{14045265501371358923053872833006846459929703}{612293656062040469012635127959717425839222518792} a^{14} + \frac{1267754209285620395615237660783771558625592469}{4898349248496323752101081023677739406713780150336} a^{13} - \frac{44850290874120020653254038393423085829986767725}{4898349248496323752101081023677739406713780150336} a^{12} + \frac{8877916793508938141151898960694704642892924575}{349882089178308839435791501691267100479555725024} a^{11} + \frac{7125796756017489462755446486370493445556482339}{699764178356617678871583003382534200959111450048} a^{10} + \frac{53097664192178423591273767473541961071272989913}{1224587312124080938025270255919434851678445037584} a^{9} + \frac{496580739480453265276671481621320533348782911}{18414846798858359970304815878487742130502932896} a^{8} - \frac{334652851212900984512681162723165821738707454529}{4898349248496323752101081023677739406713780150336} a^{7} - \frac{140271283054718616086748034712569785676078946103}{4898349248496323752101081023677739406713780150336} a^{6} + \frac{217223976324079107030122784923776345876056319855}{2449174624248161876050540511838869703356890075168} a^{5} - \frac{711688414093646122456090736864295431150097109321}{4898349248496323752101081023677739406713780150336} a^{4} + \frac{349996098370706807464542005874328764282107239129}{1399528356713235357743166006765068401918222900096} a^{3} - \frac{99797033911427425427636951880370575557771015801}{349882089178308839435791501691267100479555725024} a^{2} - \frac{220479764243877891802496187398620967242766775043}{612293656062040469012635127959717425839222518792} a - \frac{2534443630963225852391542545763422976939753138}{10933815286822151232368484427852096889986116407}$

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

Class group and class number

$C_{2}\times C_{110}\times C_{220}$, which has order $48400$ (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:  \( 15204478.8595 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{305}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{5}) \), 4.4.28372625.2, 4.4.28372625.1, \(\Q(\sqrt{5}, \sqrt{61})\), 8.0.245526784064140625.1 x2, 8.0.245526784064140625.2 x2, 8.8.805005849390625.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$61$61.8.7.2$x^{8} - 244$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61.8.7.2$x^{8} - 244$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$