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Properties

Label 16.6.73589398538...6875.2
Degree $16$
Signature $[6, 5]$
Discriminant $-\,3^{8}\cdot 5^{14}\cdot 179^{5}$
Root discriminant $35.82$
Ramified primes $3, 5, 179$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1354

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![33571, 92597, 85241, 23063, -11234, 2021, 5959, -7841, -68, 614, -876, 261, -24, -17, 16, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 16*x^14 - 17*x^13 - 24*x^12 + 261*x^11 - 876*x^10 + 614*x^9 - 68*x^8 - 7841*x^7 + 5959*x^6 + 2021*x^5 - 11234*x^4 + 23063*x^3 + 85241*x^2 + 92597*x + 33571)
 
gp: K = bnfinit(x^16 - 3*x^15 + 16*x^14 - 17*x^13 - 24*x^12 + 261*x^11 - 876*x^10 + 614*x^9 - 68*x^8 - 7841*x^7 + 5959*x^6 + 2021*x^5 - 11234*x^4 + 23063*x^3 + 85241*x^2 + 92597*x + 33571, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 16 x^{14} - 17 x^{13} - 24 x^{12} + 261 x^{11} - 876 x^{10} + 614 x^{9} - 68 x^{8} - 7841 x^{7} + 5959 x^{6} + 2021 x^{5} - 11234 x^{4} + 23063 x^{3} + 85241 x^{2} + 92597 x + 33571 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-7358939853847283935546875=-\,3^{8}\cdot 5^{14}\cdot 179^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.82$
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:  $3, 5, 179$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{6258467710797179216034526683506103} a^{15} + \frac{91455492959969662372000950278569}{6258467710797179216034526683506103} a^{14} - \frac{14066053989178798450710242035201}{106075723911816596881941130228917} a^{13} - \frac{531962091380640818351638590335938}{6258467710797179216034526683506103} a^{12} - \frac{2243816989325355634727029284251771}{6258467710797179216034526683506103} a^{11} - \frac{2357019182898218704140099962772539}{6258467710797179216034526683506103} a^{10} + \frac{838651544618705387654701355354892}{2086155903599059738678175561168701} a^{9} - \frac{2172857255415445890324783317264138}{6258467710797179216034526683506103} a^{8} - \frac{345321257401173597131383093415871}{2086155903599059738678175561168701} a^{7} + \frac{2938802782258093731826329216592787}{6258467710797179216034526683506103} a^{6} + \frac{981886568910734683678953933547678}{2086155903599059738678175561168701} a^{5} + \frac{1510147551096341254143467077603199}{6258467710797179216034526683506103} a^{4} - \frac{352764463229662192370592983953889}{2086155903599059738678175561168701} a^{3} - \frac{330583710169339176299047159653592}{6258467710797179216034526683506103} a^{2} - \frac{1774124994676525466506962059971055}{6258467710797179216034526683506103} a + \frac{3199086833517787718490125666221}{35358574637272198960647043409639}$

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:  $10$
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:  \( 1992298.09627 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1354:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 59 conjugacy class representatives for t16n1354 are not computed
Character table for t16n1354 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.6.226546875.1

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 $16$ R R $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
179Data not computed