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Properties

Label 16.8.27404580206...7024.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{30}\cdot 761^{5}$
Root discriminant $29.17$
Ramified primes $2, 761$
Class number $1$
Class group Trivial
Galois group 16T1870

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![46, -68, -497, 838, 767, -1480, -217, 1226, -837, 272, 12, 4, -66, 32, 5, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 5*x^14 + 32*x^13 - 66*x^12 + 4*x^11 + 12*x^10 + 272*x^9 - 837*x^8 + 1226*x^7 - 217*x^6 - 1480*x^5 + 767*x^4 + 838*x^3 - 497*x^2 - 68*x + 46)
 
gp: K = bnfinit(x^16 - 6*x^15 + 5*x^14 + 32*x^13 - 66*x^12 + 4*x^11 + 12*x^10 + 272*x^9 - 837*x^8 + 1226*x^7 - 217*x^6 - 1480*x^5 + 767*x^4 + 838*x^3 - 497*x^2 - 68*x + 46, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 5 x^{14} + 32 x^{13} - 66 x^{12} + 4 x^{11} + 12 x^{10} + 272 x^{9} - 837 x^{8} + 1226 x^{7} - 217 x^{6} - 1480 x^{5} + 767 x^{4} + 838 x^{3} - 497 x^{2} - 68 x + 46 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(274045802062918423937024=2^{30}\cdot 761^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.17$
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:  $2, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{15} a^{13} - \frac{1}{15} a^{12} + \frac{1}{15} a^{11} - \frac{7}{15} a^{10} - \frac{2}{5} a^{9} - \frac{4}{15} a^{8} - \frac{7}{15} a^{7} - \frac{1}{15} a^{6} + \frac{7}{15} a^{5} + \frac{7}{15} a^{4} + \frac{1}{5} a^{3} - \frac{7}{15} a^{2} + \frac{1}{3} a - \frac{2}{15}$, $\frac{1}{75} a^{14} + \frac{2}{75} a^{13} - \frac{2}{75} a^{12} + \frac{1}{15} a^{11} - \frac{2}{5} a^{10} + \frac{29}{75} a^{9} - \frac{1}{75} a^{8} + \frac{17}{75} a^{7} + \frac{22}{75} a^{6} + \frac{1}{75} a^{5} - \frac{13}{75} a^{3} - \frac{7}{75} a^{2} - \frac{29}{75} a - \frac{1}{25}$, $\frac{1}{11245133087256900} a^{15} + \frac{2704611432689}{3748377695752300} a^{14} - \frac{13693372053731}{937094423938075} a^{13} - \frac{18163471121401}{187418884787615} a^{12} - \frac{110267527057093}{1124513308725690} a^{11} + \frac{618118770704677}{5622566543628450} a^{10} + \frac{2385245050197017}{5622566543628450} a^{9} - \frac{2369419651100299}{5622566543628450} a^{8} + \frac{255994919972679}{3748377695752300} a^{7} + \frac{2631047674347671}{11245133087256900} a^{6} + \frac{444446240030101}{1124513308725690} a^{5} - \frac{89204368733759}{5622566543628450} a^{4} + \frac{3128622075039913}{11245133087256900} a^{3} - \frac{875355215478583}{3748377695752300} a^{2} + \frac{1624776598389401}{5622566543628450} a + \frac{185030092891273}{374837769575230}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2204961.39622 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1870:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 83 conjugacy class representatives for t16n1870 are not computed
Character table for t16n1870 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 8.4.2372079616.2

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.12.24.244$x^{12} - 8 x^{11} + 4 x^{10} - 8 x^{9} - 14 x^{8} + 8 x^{7} + 8 x^{5} + 16 x^{3} - 8 x^{2} + 16 x + 8$$4$$3$$24$$C_2^2 \times A_4$$[2, 2, 3]^{6}$
761Data not computed