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Properties

Label 16.4.15673665258...6729.1
Degree $16$
Signature $[4, 6]$
Discriminant $3^{10}\cdot 61^{12}$
Root discriminant $43.37$
Ramified primes $3, 61$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2^4.D_4$ (as 16T330)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9225, 59820, 23524, 46998, 58619, -6799, 1233, 1069, -2805, -405, 231, -359, 99, -40, 14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 14*x^14 - 40*x^13 + 99*x^12 - 359*x^11 + 231*x^10 - 405*x^9 - 2805*x^8 + 1069*x^7 + 1233*x^6 - 6799*x^5 + 58619*x^4 + 46998*x^3 + 23524*x^2 + 59820*x - 9225)
 
gp: K = bnfinit(x^16 + 14*x^14 - 40*x^13 + 99*x^12 - 359*x^11 + 231*x^10 - 405*x^9 - 2805*x^8 + 1069*x^7 + 1233*x^6 - 6799*x^5 + 58619*x^4 + 46998*x^3 + 23524*x^2 + 59820*x - 9225, 1)
 

Normalized defining polynomial

\( x^{16} + 14 x^{14} - 40 x^{13} + 99 x^{12} - 359 x^{11} + 231 x^{10} - 405 x^{9} - 2805 x^{8} + 1069 x^{7} + 1233 x^{6} - 6799 x^{5} + 58619 x^{4} + 46998 x^{3} + 23524 x^{2} + 59820 x - 9225 \)

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:  \(156736652583298165062696729=3^{10}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{4} + \frac{2}{5} a$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{5} + \frac{2}{5} a^{2}$, $\frac{1}{15} a^{9} - \frac{1}{15} a^{7} - \frac{2}{5} a^{5} - \frac{1}{15} a^{3} + \frac{2}{5} a^{2} + \frac{1}{15} a$, $\frac{1}{15} a^{10} - \frac{1}{15} a^{8} + \frac{1}{5} a^{5} - \frac{7}{15} a^{4} - \frac{2}{15} a^{2} + \frac{2}{5} a$, $\frac{1}{75} a^{11} + \frac{1}{75} a^{10} + \frac{2}{75} a^{9} + \frac{2}{75} a^{8} - \frac{2}{25} a^{7} + \frac{2}{25} a^{6} - \frac{37}{75} a^{5} + \frac{14}{75} a^{4} - \frac{23}{75} a^{3} + \frac{34}{75} a^{2} - \frac{3}{25} a + \frac{1}{5}$, $\frac{1}{225} a^{12} - \frac{1}{225} a^{11} - \frac{2}{225} a^{9} - \frac{2}{45} a^{8} + \frac{1}{75} a^{7} - \frac{19}{225} a^{6} + \frac{28}{225} a^{5} + \frac{13}{75} a^{4} - \frac{4}{9} a^{3} + \frac{58}{225} a^{2} + \frac{11}{75} a + \frac{1}{5}$, $\frac{1}{225} a^{13} - \frac{1}{225} a^{11} - \frac{2}{225} a^{10} + \frac{1}{75} a^{9} - \frac{7}{225} a^{8} + \frac{14}{225} a^{7} + \frac{1}{25} a^{6} - \frac{23}{225} a^{5} + \frac{29}{225} a^{4} - \frac{19}{75} a^{3} - \frac{44}{225} a^{2} - \frac{14}{75} a + \frac{1}{5}$, $\frac{1}{675} a^{14} + \frac{1}{675} a^{13} - \frac{1}{675} a^{12} - \frac{1}{225} a^{11} - \frac{14}{675} a^{10} - \frac{19}{675} a^{9} + \frac{67}{675} a^{8} - \frac{52}{675} a^{7} - \frac{59}{675} a^{6} + \frac{77}{225} a^{5} - \frac{58}{675} a^{4} + \frac{184}{675} a^{3} + \frac{79}{675} a^{2} - \frac{34}{225} a - \frac{4}{15}$, $\frac{1}{393368406827609451722998275} a^{15} + \frac{67856325138963746489929}{393368406827609451722998275} a^{14} - \frac{13615535286988020680831}{78673681365521890344599655} a^{13} - \frac{94441488236523643682177}{131122802275869817240999425} a^{12} - \frac{862460615996277763074083}{393368406827609451722998275} a^{11} - \frac{6045817015283745482147299}{393368406827609451722998275} a^{10} - \frac{2431962763366202210108392}{78673681365521890344599655} a^{9} - \frac{31342368700776025925625661}{393368406827609451722998275} a^{8} - \frac{17484699975856579833893966}{393368406827609451722998275} a^{7} + \frac{394629370685890980049271}{43707600758623272413666475} a^{6} + \frac{8433088812432682874450098}{78673681365521890344599655} a^{5} - \frac{18575435590724058920645726}{393368406827609451722998275} a^{4} + \frac{4741789388591978090533261}{393368406827609451722998275} a^{3} + \frac{1114513487397523953242236}{5244912091034792689639977} a^{2} - \frac{8420776171608595968771206}{43707600758623272413666475} a + \frac{437304307451920589143114}{2913840050574884827577765}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 5402015.56141 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.D_4$ (as 16T330):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4.D_4$
Character table for $C_2^4.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{61}) \), 4.2.11163.1, 8.2.1391050107747.2, 8.4.68412300381.1, 8.2.205236901143.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
61Data not computed