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Properties

Label 16.4.37038789304...0625.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 79^{8}$
Root discriminant $29.72$
Ramified primes $5, 79$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_4:C_4$ (as 16T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1999, 505, 6964, -1264, -9488, 3963, 4913, -2765, -1788, 1571, -18, -229, 16, 17, 7, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 7*x^14 + 17*x^13 + 16*x^12 - 229*x^11 - 18*x^10 + 1571*x^9 - 1788*x^8 - 2765*x^7 + 4913*x^6 + 3963*x^5 - 9488*x^4 - 1264*x^3 + 6964*x^2 + 505*x - 1999)
 
gp: K = bnfinit(x^16 - 6*x^15 + 7*x^14 + 17*x^13 + 16*x^12 - 229*x^11 - 18*x^10 + 1571*x^9 - 1788*x^8 - 2765*x^7 + 4913*x^6 + 3963*x^5 - 9488*x^4 - 1264*x^3 + 6964*x^2 + 505*x - 1999, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 7 x^{14} + 17 x^{13} + 16 x^{12} - 229 x^{11} - 18 x^{10} + 1571 x^{9} - 1788 x^{8} - 2765 x^{7} + 4913 x^{6} + 3963 x^{5} - 9488 x^{4} - 1264 x^{3} + 6964 x^{2} + 505 x - 1999 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(370387893043593994140625=5^{12}\cdot 79^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.72$
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:  $5, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{10} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{3}{10} a^{5} + \frac{1}{10} a^{4} - \frac{3}{10} a^{3} - \frac{1}{10} a^{2} - \frac{1}{10} a + \frac{1}{10}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{11} + \frac{1}{10} a^{10} - \frac{1}{10} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{10} a^{4} + \frac{3}{10} a^{3} - \frac{3}{10} a^{2} + \frac{2}{5} a - \frac{3}{10}$, $\frac{1}{20} a^{14} - \frac{1}{20} a^{11} + \frac{1}{10} a^{10} + \frac{1}{5} a^{9} - \frac{9}{20} a^{7} + \frac{1}{5} a^{6} + \frac{3}{10} a^{5} - \frac{1}{20} a^{4} - \frac{1}{20} a^{3} - \frac{7}{20} a^{2} - \frac{9}{20} a - \frac{9}{20}$, $\frac{1}{1020039091315264669640} a^{15} + \frac{24751995309510499723}{1020039091315264669640} a^{14} - \frac{4063948496976419311}{510019545657632334820} a^{13} + \frac{28599971941275814371}{1020039091315264669640} a^{12} + \frac{25560264739791879067}{1020039091315264669640} a^{11} + \frac{107084994063671929057}{510019545657632334820} a^{10} + \frac{20219216497935616748}{127504886414408083705} a^{9} + \frac{387731997480341337531}{1020039091315264669640} a^{8} - \frac{184711387995980887533}{1020039091315264669640} a^{7} + \frac{101130684807270553833}{510019545657632334820} a^{6} - \frac{462487535631578645929}{1020039091315264669640} a^{5} - \frac{12468077430995566361}{102003909131526466964} a^{4} - \frac{217625894969234529171}{510019545657632334820} a^{3} + \frac{71591130748709115127}{510019545657632334820} a^{2} + \frac{44466019485286240917}{510019545657632334820} a + \frac{230096886064827028931}{1020039091315264669640}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $9$
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:  \( 306662.895205 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.780125.1, 4.2.1975.1, 4.2.9875.1, 8.2.7703734375.1, 8.2.308149375.1, 8.4.608595015625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
79Data not computed