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Properties

Label 16.0.15905028274...6289.1
Degree $16$
Signature $[0, 8]$
Discriminant $19^{4}\cdot 73^{14}$
Root discriminant $89.14$
Ramified primes $19, 73$
Class number $89$ (GRH)
Class group $[89]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1600672, -6972784, 13679510, -15825337, 11891245, -5981569, 2003583, -434233, 61895, -7010, 164, 173, 163, -101, 15, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 15*x^14 - 101*x^13 + 163*x^12 + 173*x^11 + 164*x^10 - 7010*x^9 + 61895*x^8 - 434233*x^7 + 2003583*x^6 - 5981569*x^5 + 11891245*x^4 - 15825337*x^3 + 13679510*x^2 - 6972784*x + 1600672)
 
gp: K = bnfinit(x^16 - 3*x^15 + 15*x^14 - 101*x^13 + 163*x^12 + 173*x^11 + 164*x^10 - 7010*x^9 + 61895*x^8 - 434233*x^7 + 2003583*x^6 - 5981569*x^5 + 11891245*x^4 - 15825337*x^3 + 13679510*x^2 - 6972784*x + 1600672, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 15 x^{14} - 101 x^{13} + 163 x^{12} + 173 x^{11} + 164 x^{10} - 7010 x^{9} + 61895 x^{8} - 434233 x^{7} + 2003583 x^{6} - 5981569 x^{5} + 11891245 x^{4} - 15825337 x^{3} + 13679510 x^{2} - 6972784 x + 1600672 \)

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:  \(15905028274673815182544659396289=19^{4}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.14$
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:  $19, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{10} + \frac{1}{16} a^{8} - \frac{1}{8} a^{5} + \frac{1}{16} a^{4} - \frac{1}{2} a^{3} - \frac{3}{16} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{32} a^{11} + \frac{1}{32} a^{9} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} + \frac{5}{32} a^{5} - \frac{1}{8} a^{4} - \frac{3}{32} a^{3} + \frac{3}{16} a^{2}$, $\frac{1}{64} a^{12} - \frac{1}{64} a^{11} + \frac{1}{64} a^{10} - \frac{1}{64} a^{9} - \frac{1}{16} a^{8} + \frac{1}{32} a^{7} + \frac{7}{64} a^{6} - \frac{9}{64} a^{5} + \frac{1}{64} a^{4} - \frac{23}{64} a^{3} + \frac{13}{32} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{512} a^{13} + \frac{1}{64} a^{10} - \frac{5}{512} a^{9} + \frac{19}{256} a^{8} - \frac{55}{512} a^{7} + \frac{15}{256} a^{6} + \frac{9}{64} a^{5} - \frac{7}{256} a^{4} + \frac{163}{512} a^{3} - \frac{15}{256} a^{2} + \frac{5}{32} a + \frac{7}{16}$, $\frac{1}{1024} a^{14} - \frac{1}{1024} a^{13} + \frac{1}{128} a^{11} - \frac{13}{1024} a^{10} + \frac{43}{1024} a^{9} - \frac{93}{1024} a^{8} - \frac{43}{1024} a^{7} + \frac{21}{512} a^{6} - \frac{107}{512} a^{5} - \frac{207}{1024} a^{4} + \frac{319}{1024} a^{3} + \frac{247}{512} a^{2} + \frac{25}{64} a + \frac{9}{32}$, $\frac{1}{80893646556392978631541694464} a^{15} - \frac{1807187763668061760684783}{5055852909774561164471355904} a^{14} + \frac{27517873583900449756432319}{80893646556392978631541694464} a^{13} + \frac{29351670060509633760190161}{10111705819549122328942711808} a^{12} + \frac{1185984623237000748521743035}{80893646556392978631541694464} a^{11} + \frac{598748797542652878399800775}{40446823278196489315770847232} a^{10} + \frac{2290465925777685089581322583}{40446823278196489315770847232} a^{9} + \frac{845149808900871350095263065}{10111705819549122328942711808} a^{8} + \frac{7490765794570650176091347103}{80893646556392978631541694464} a^{7} + \frac{1608388482058351493892933781}{20223411639098244657885423616} a^{6} - \frac{15697535968429376238861935813}{80893646556392978631541694464} a^{5} + \frac{310138137946987589783341151}{2527926454887280582235677952} a^{4} - \frac{23885193808257096503445340211}{80893646556392978631541694464} a^{3} - \frac{19004788810982243412779306913}{40446823278196489315770847232} a^{2} - \frac{1932501082012881190035731213}{5055852909774561164471355904} a + \frac{291833753278543787700216345}{2527926454887280582235677952}$

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

Class group and class number

$C_{89}$, which has order $89$ (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:  \( 4629594070.98 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 4.2.7391323.1, 4.2.101251.1, 8.4.3988110865394017.1, 8.0.11047398519097.1, 8.4.54631655690329.1

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 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$19$19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
73Data not computed