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Properties

Label 16.0.24722989956...4209.4
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 59^{8}$
Root discriminant $91.64$
Ramified primes $17, 59$
Class number $24$ (GRH)
Class group $[24]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8353648, 1464552, -748608, -1158746, -47635, 602240, 31514, -164840, 2364, 19994, 1466, -2440, -18, 168, -4, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 4*x^14 + 168*x^13 - 18*x^12 - 2440*x^11 + 1466*x^10 + 19994*x^9 + 2364*x^8 - 164840*x^7 + 31514*x^6 + 602240*x^5 - 47635*x^4 - 1158746*x^3 - 748608*x^2 + 1464552*x + 8353648)
 
gp: K = bnfinit(x^16 - 8*x^15 - 4*x^14 + 168*x^13 - 18*x^12 - 2440*x^11 + 1466*x^10 + 19994*x^9 + 2364*x^8 - 164840*x^7 + 31514*x^6 + 602240*x^5 - 47635*x^4 - 1158746*x^3 - 748608*x^2 + 1464552*x + 8353648, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 4 x^{14} + 168 x^{13} - 18 x^{12} - 2440 x^{11} + 1466 x^{10} + 19994 x^{9} + 2364 x^{8} - 164840 x^{7} + 31514 x^{6} + 602240 x^{5} - 47635 x^{4} - 1158746 x^{3} - 748608 x^{2} + 1464552 x + 8353648 \)

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:  \(24722989956581301387479701814209=17^{14}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.64$
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:  $17, 59$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{68} a^{8} - \frac{1}{17} a^{7} - \frac{5}{34} a^{6} + \frac{5}{34} a^{5} - \frac{1}{34} a^{4} - \frac{3}{34} a^{3} - \frac{7}{68} a^{2} + \frac{9}{34} a - \frac{4}{17}$, $\frac{1}{68} a^{9} + \frac{2}{17} a^{7} + \frac{1}{17} a^{6} + \frac{1}{17} a^{5} - \frac{7}{34} a^{4} + \frac{3}{68} a^{3} + \frac{6}{17} a^{2} + \frac{11}{34} a + \frac{1}{17}$, $\frac{1}{68} a^{10} + \frac{1}{34} a^{7} + \frac{4}{17} a^{6} + \frac{2}{17} a^{5} - \frac{15}{68} a^{4} + \frac{1}{17} a^{3} - \frac{6}{17} a^{2} - \frac{1}{17} a - \frac{2}{17}$, $\frac{1}{68} a^{11} - \frac{5}{34} a^{7} - \frac{3}{34} a^{6} - \frac{1}{68} a^{5} + \frac{2}{17} a^{4} + \frac{11}{34} a^{3} - \frac{6}{17} a^{2} - \frac{5}{34} a + \frac{8}{17}$, $\frac{1}{1361528368} a^{12} - \frac{3}{680764184} a^{11} + \frac{55940}{85095523} a^{10} - \frac{4475145}{1361528368} a^{9} + \frac{1507}{85095523} a^{8} + \frac{6688589}{340382092} a^{7} - \frac{321564861}{1361528368} a^{6} + \frac{85747305}{680764184} a^{5} - \frac{1605901}{340382092} a^{4} - \frac{5892075}{1361528368} a^{3} - \frac{60216441}{340382092} a^{2} + \frac{47506667}{170191046} a - \frac{569927}{170191046}$, $\frac{1}{1361528368} a^{13} + \frac{223751}{340382092} a^{11} + \frac{895095}{1361528368} a^{10} - \frac{3402141}{680764184} a^{9} + \frac{859569}{170191046} a^{8} + \frac{79230987}{1361528368} a^{7} - \frac{29012881}{340382092} a^{6} + \frac{27705627}{170191046} a^{5} - \frac{284703411}{1361528368} a^{4} - \frac{48007965}{680764184} a^{3} + \frac{59079923}{340382092} a^{2} - \frac{45900779}{170191046} a + \frac{23318314}{85095523}$, $\frac{1}{1361528368} a^{14} + \frac{6265119}{1361528368} a^{11} + \frac{4028921}{680764184} a^{10} + \frac{217223}{85095523} a^{9} + \frac{3033763}{1361528368} a^{8} - \frac{23047757}{340382092} a^{7} + \frac{17422291}{340382092} a^{6} - \frac{310066099}{1361528368} a^{5} - \frac{69216363}{680764184} a^{4} + \frac{101894075}{340382092} a^{3} - \frac{77175285}{170191046} a^{2} + \frac{9221190}{85095523} a + \frac{42223637}{85095523}$, $\frac{1}{60714634514224} a^{15} + \frac{22289}{60714634514224} a^{14} - \frac{2789}{15178658628556} a^{13} - \frac{2705}{30357317257112} a^{12} - \frac{366452701497}{60714634514224} a^{11} + \frac{107507912861}{30357317257112} a^{10} + \frac{8711369307}{7589329314278} a^{9} + \frac{22998320637}{3195507079696} a^{8} + \frac{576615903997}{3794664657139} a^{7} - \frac{1790296669629}{30357317257112} a^{6} - \frac{12532958676611}{60714634514224} a^{5} + \frac{2814364035929}{30357317257112} a^{4} + \frac{10257132477203}{60714634514224} a^{3} - \frac{6611517114389}{15178658628556} a^{2} + \frac{99639391544}{223215568067} a - \frac{3653440803737}{7589329314278}$

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

Class group and class number

$C_{24}$, which has order $24$ (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:  \( 335717402.221 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-59}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-1003}) \), \(\Q(\sqrt{17}, \sqrt{-59})\), 4.2.289867.1 x2, 4.0.17102153.1 x2, 8.0.292483637235409.1, 8.2.84274946322067.1 x4

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ R

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$17.8.7.2$x^{8} - 153$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.2$x^{8} - 153$$8$$1$$7$$C_8$$[\ ]_{8}$
$59$59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$