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Properties

Label 16.2.14501696984...2159.7
Degree $16$
Signature $[2, 7]$
Discriminant $-\,17^{15}\cdot 47^{7}$
Root discriminant $76.75$
Ramified primes $17, 47$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3775, -561890, 1073416, -1180866, 782497, -275084, 67736, -46509, 19736, -7406, 3279, -226, 302, -27, 26, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 26*x^14 - 27*x^13 + 302*x^12 - 226*x^11 + 3279*x^10 - 7406*x^9 + 19736*x^8 - 46509*x^7 + 67736*x^6 - 275084*x^5 + 782497*x^4 - 1180866*x^3 + 1073416*x^2 - 561890*x + 3775)
 
gp: K = bnfinit(x^16 - 3*x^15 + 26*x^14 - 27*x^13 + 302*x^12 - 226*x^11 + 3279*x^10 - 7406*x^9 + 19736*x^8 - 46509*x^7 + 67736*x^6 - 275084*x^5 + 782497*x^4 - 1180866*x^3 + 1073416*x^2 - 561890*x + 3775, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 26 x^{14} - 27 x^{13} + 302 x^{12} - 226 x^{11} + 3279 x^{10} - 7406 x^{9} + 19736 x^{8} - 46509 x^{7} + 67736 x^{6} - 275084 x^{5} + 782497 x^{4} - 1180866 x^{3} + 1073416 x^{2} - 561890 x + 3775 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1450169698441125460523978872159=-\,17^{15}\cdot 47^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.75$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{10} a^{12} - \frac{1}{2} a^{9} - \frac{1}{5} a^{7} + \frac{3}{10} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{10} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{1}{2}$, $\frac{1}{50} a^{13} - \frac{1}{25} a^{12} + \frac{1}{50} a^{10} + \frac{4}{25} a^{9} + \frac{4}{25} a^{8} + \frac{3}{10} a^{7} + \frac{1}{25} a^{6} + \frac{9}{25} a^{5} + \frac{1}{50} a^{4} + \frac{2}{5} a^{3} - \frac{9}{25} a^{2} - \frac{11}{50} a + \frac{2}{5}$, $\frac{1}{5350} a^{14} + \frac{19}{5350} a^{13} - \frac{147}{5350} a^{12} - \frac{259}{5350} a^{11} - \frac{481}{5350} a^{10} - \frac{2189}{5350} a^{9} + \frac{1103}{5350} a^{8} - \frac{1903}{5350} a^{7} + \frac{319}{1070} a^{6} + \frac{2059}{5350} a^{5} + \frac{2421}{5350} a^{4} + \frac{1957}{5350} a^{3} + \frac{1751}{5350} a^{2} + \frac{2639}{5350} a + \frac{409}{1070}$, $\frac{1}{15976577110391531339230168924947037250} a^{15} + \frac{7647597385524017929461637433842}{84087247949429112311737731183931775} a^{14} - \frac{862309055354914948961247660235529}{1452416100944684667202742629540639750} a^{13} - \frac{336258241899569424132465043294877977}{7988288555195765669615084462473518625} a^{12} - \frac{98797166117249922091234964400003364}{1597657711039153133923016892494703725} a^{11} + \frac{5390998518472654250768042963294679}{1452416100944684667202742629540639750} a^{10} + \frac{3158977245853677799420451108017467378}{7988288555195765669615084462473518625} a^{9} + \frac{46737934458990114617473593383227601}{7988288555195765669615084462473518625} a^{8} - \frac{14341134024970489925154862969042983}{15976577110391531339230168924947037250} a^{7} + \frac{63976892739719941364408427573304966}{7988288555195765669615084462473518625} a^{6} + \frac{507531803046564301675254531245323461}{7988288555195765669615084462473518625} a^{5} - \frac{702795540394274019554105416886855373}{1452416100944684667202742629540639750} a^{4} + \frac{2202194473372371933206795615762387214}{7988288555195765669615084462473518625} a^{3} + \frac{3096813359282830903002800078135816594}{7988288555195765669615084462473518625} a^{2} - \frac{1435647514561966834019652731556863177}{3195315422078306267846033784989407450} a + \frac{35674905695928061130442074234909349}{127812616883132250713841351399576298}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  $8$
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:  \( 70490191.6085 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

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 $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.2.230911.1, 8.2.42602592046879.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
17Data not computed
47Data not computed