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Properties

Label 16.0.17415183620...4081.4
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 61^{12}$
Root discriminant $37.81$
Ramified primes $3, 61$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28561, -19773, 12337, -17771, 12297, -5520, 1659, -849, 832, -187, -107, 30, 41, -36, 15, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 15*x^14 - 36*x^13 + 41*x^12 + 30*x^11 - 107*x^10 - 187*x^9 + 832*x^8 - 849*x^7 + 1659*x^6 - 5520*x^5 + 12297*x^4 - 17771*x^3 + 12337*x^2 - 19773*x + 28561)
 
gp: K = bnfinit(x^16 - 5*x^15 + 15*x^14 - 36*x^13 + 41*x^12 + 30*x^11 - 107*x^10 - 187*x^9 + 832*x^8 - 849*x^7 + 1659*x^6 - 5520*x^5 + 12297*x^4 - 17771*x^3 + 12337*x^2 - 19773*x + 28561, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 15 x^{14} - 36 x^{13} + 41 x^{12} + 30 x^{11} - 107 x^{10} - 187 x^{9} + 832 x^{8} - 849 x^{7} + 1659 x^{6} - 5520 x^{5} + 12297 x^{4} - 17771 x^{3} + 12337 x^{2} - 19773 x + 28561 \)

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:  \(17415183620366462784744081=3^{8}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.81$
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:  $3, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{20} a^{12} - \frac{1}{5} a^{11} + \frac{1}{10} a^{10} + \frac{1}{4} a^{9} - \frac{3}{20} a^{8} + \frac{1}{5} a^{7} + \frac{1}{20} a^{6} - \frac{3}{10} a^{5} + \frac{3}{10} a^{4} - \frac{1}{10} a^{3} + \frac{3}{10} a^{2} - \frac{1}{5} a - \frac{1}{20}$, $\frac{1}{780} a^{13} - \frac{3}{130} a^{12} + \frac{79}{390} a^{11} + \frac{79}{260} a^{10} + \frac{109}{260} a^{9} - \frac{37}{390} a^{8} - \frac{21}{52} a^{7} - \frac{10}{39} a^{6} - \frac{1}{6} a^{5} - \frac{31}{130} a^{4} + \frac{49}{130} a^{3} - \frac{2}{195} a^{2} + \frac{31}{156} a - \frac{1}{30}$, $\frac{1}{3701100} a^{14} + \frac{57}{308425} a^{13} + \frac{13592}{925275} a^{12} + \frac{106523}{1233700} a^{11} + \frac{89991}{1233700} a^{10} - \frac{164837}{370110} a^{9} - \frac{294867}{1233700} a^{8} - \frac{429373}{1850550} a^{7} - \frac{18442}{71175} a^{6} - \frac{6817}{616850} a^{5} + \frac{257683}{616850} a^{4} + \frac{1264}{12675} a^{3} + \frac{36167}{148044} a^{2} + \frac{979}{2847} a + \frac{103}{3650}$, $\frac{1}{1115873928887312598900} a^{15} + \frac{6274260179983}{55793696444365629945} a^{14} + \frac{255736755257264777}{1115873928887312598900} a^{13} - \frac{13574811975665870389}{557936964443656299450} a^{12} - \frac{55470714222613323291}{371957976295770866300} a^{11} - \frac{470488250932268717027}{1115873928887312598900} a^{10} - \frac{32596934011179725671}{1115873928887312598900} a^{9} + \frac{105088665519167434723}{1115873928887312598900} a^{8} - \frac{8317001959028699011}{17167291213650963060} a^{7} + \frac{60999806099611547749}{1115873928887312598900} a^{6} + \frac{31918284027866914181}{185978988147885433150} a^{5} - \frac{100092505605507349976}{278968482221828149725} a^{4} + \frac{245418172914048894343}{1115873928887312598900} a^{3} - \frac{1076710476234536653}{8583645606825481530} a^{2} - \frac{1813008458286657367}{6602804312942678100} a - \frac{9174625116446523}{169302674690837900}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 615708.74713 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{61}) \), 4.2.11163.1, 4.0.226981.1, 4.2.680943.1, 8.0.68412300381.1, 8.4.68412300381.1, 8.0.463683369249.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$61$61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$