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Properties

Label 16.0.77190325834...9953.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{11}\cdot 83^{8}$
Root discriminant $63.90$
Ramified primes $17, 83$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5689003, -2274735, 1544858, -718066, 388293, -65293, 13639, 17194, -454, -2227, 2250, -837, 245, -43, 19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 19*x^14 - 43*x^13 + 245*x^12 - 837*x^11 + 2250*x^10 - 2227*x^9 - 454*x^8 + 17194*x^7 + 13639*x^6 - 65293*x^5 + 388293*x^4 - 718066*x^3 + 1544858*x^2 - 2274735*x + 5689003)
 
gp: K = bnfinit(x^16 - 6*x^15 + 19*x^14 - 43*x^13 + 245*x^12 - 837*x^11 + 2250*x^10 - 2227*x^9 - 454*x^8 + 17194*x^7 + 13639*x^6 - 65293*x^5 + 388293*x^4 - 718066*x^3 + 1544858*x^2 - 2274735*x + 5689003, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 19 x^{14} - 43 x^{13} + 245 x^{12} - 837 x^{11} + 2250 x^{10} - 2227 x^{9} - 454 x^{8} + 17194 x^{7} + 13639 x^{6} - 65293 x^{5} + 388293 x^{4} - 718066 x^{3} + 1544858 x^{2} - 2274735 x + 5689003 \)

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:  \(77190325834356486941365599953=17^{11}\cdot 83^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.90$
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:  $17, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{11} + \frac{1}{7} a^{10} + \frac{1}{7} a^{9} - \frac{3}{7} a^{8} - \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3} + \frac{3}{7} a^{2} + \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{13} - \frac{1}{7} a^{11} - \frac{2}{7} a^{10} + \frac{1}{7} a^{9} + \frac{2}{7} a^{8} - \frac{1}{7} a^{7} + \frac{3}{7} a^{5} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{147} a^{14} - \frac{5}{147} a^{13} + \frac{2}{49} a^{12} + \frac{1}{49} a^{11} + \frac{53}{147} a^{10} - \frac{73}{147} a^{9} - \frac{11}{147} a^{8} - \frac{37}{147} a^{7} + \frac{38}{147} a^{6} + \frac{5}{21} a^{5} - \frac{1}{21} a^{4} - \frac{6}{49} a^{3} - \frac{43}{147} a^{2} - \frac{17}{147} a + \frac{68}{147}$, $\frac{1}{45147411445920705563835677235403355593851549} a^{15} + \frac{43820359931927361325191634370634794095480}{45147411445920705563835677235403355593851549} a^{14} + \frac{45733276722714654815192134883086797319927}{2149876735520033598277889392162064552088169} a^{13} - \frac{492682559313027474210974945166991161030146}{15049137148640235187945225745134451864617183} a^{12} + \frac{10530228250631877879273022351222229484210344}{45147411445920705563835677235403355593851549} a^{11} + \frac{2074604521511045489742344535320479008511964}{45147411445920705563835677235403355593851549} a^{10} + \frac{13750245402497781186201625798296920712734638}{45147411445920705563835677235403355593851549} a^{9} - \frac{4316996455795748750579312254168097374874635}{45147411445920705563835677235403355593851549} a^{8} + \frac{157975648231548246732720084421831967203708}{1962930932431335024514594662408841547558763} a^{7} + \frac{12049386647932843333801180698559740969817445}{45147411445920705563835677235403355593851549} a^{6} + \frac{3175021137681648018829696561178023984795415}{6449630206560100794833668176486193656264507} a^{5} - \frac{5747852127435483796980465552281871094124409}{15049137148640235187945225745134451864617183} a^{4} - \frac{20775938464300515356460576680580686836677325}{45147411445920705563835677235403355593851549} a^{3} - \frac{120186105803715936221854472648751490911551}{921375743794300113547666882355170522323501} a^{2} + \frac{5162935900596119809704333461261533185514}{40059814947578265806420299232833500970587} a + \frac{6867389571080811923486565196724490096256063}{15049137148640235187945225745134451864617183}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 24027892.3072 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-83}) \), 4.0.117113.1, 8.0.233162731073.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$17$17.8.7.5$x^{8} + 459$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.4.2$x^{8} - 4913 x^{2} + 918731$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
83Data not computed