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Properties

Label 16.0.37909967031...8041.5
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 61^{10}$
Root discriminant $29.76$
Ramified primes $3, 61$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^2.SD_{16}$ (as 16T163)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![53971, -23830, 35864, -19272, -2581, 14875, -4016, -3528, 4567, -2937, 1144, -418, 191, -84, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 191*x^12 - 418*x^11 + 1144*x^10 - 2937*x^9 + 4567*x^8 - 3528*x^7 - 4016*x^6 + 14875*x^5 - 2581*x^4 - 19272*x^3 + 35864*x^2 - 23830*x + 53971)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 191*x^12 - 418*x^11 + 1144*x^10 - 2937*x^9 + 4567*x^8 - 3528*x^7 - 4016*x^6 + 14875*x^5 - 2581*x^4 - 19272*x^3 + 35864*x^2 - 23830*x + 53971, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 191 x^{12} - 418 x^{11} + 1144 x^{10} - 2937 x^{9} + 4567 x^{8} - 3528 x^{7} - 4016 x^{6} + 14875 x^{5} - 2581 x^{4} - 19272 x^{3} + 35864 x^{2} - 23830 x + 53971 \)

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:  \(379099670317033992358041=3^{12}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.76$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{15} a^{8} - \frac{1}{15} a^{7} - \frac{1}{15} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{7}{15} a^{3} + \frac{2}{15} a^{2} + \frac{1}{15} a - \frac{1}{3}$, $\frac{1}{15} a^{9} + \frac{1}{15} a^{7} + \frac{1}{15} a^{6} - \frac{1}{5} a^{5} + \frac{2}{15} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{15} a - \frac{2}{15}$, $\frac{1}{75} a^{10} - \frac{1}{75} a^{8} - \frac{2}{25} a^{7} - \frac{37}{75} a^{6} + \frac{1}{75} a^{5} - \frac{11}{75} a^{4} + \frac{2}{15} a^{3} - \frac{7}{15} a^{2} - \frac{22}{75} a + \frac{16}{75}$, $\frac{1}{75} a^{11} - \frac{1}{75} a^{9} - \frac{1}{75} a^{8} + \frac{1}{25} a^{7} + \frac{26}{75} a^{6} - \frac{31}{75} a^{5} - \frac{1}{5} a^{4} + \frac{1}{25} a^{2} + \frac{12}{25} a + \frac{4}{15}$, $\frac{1}{375} a^{12} - \frac{1}{375} a^{11} + \frac{2}{375} a^{10} - \frac{2}{75} a^{9} + \frac{1}{375} a^{8} - \frac{4}{75} a^{7} + \frac{62}{375} a^{6} + \frac{64}{375} a^{5} + \frac{112}{375} a^{4} - \frac{19}{125} a^{3} - \frac{14}{125} a^{2} + \frac{6}{125} a + \frac{36}{125}$, $\frac{1}{375} a^{13} + \frac{1}{375} a^{11} + \frac{2}{375} a^{10} - \frac{3}{125} a^{9} - \frac{4}{375} a^{8} + \frac{32}{375} a^{7} + \frac{31}{375} a^{6} - \frac{139}{375} a^{5} - \frac{12}{25} a^{4} + \frac{176}{375} a^{3} - \frac{58}{125} a^{2} + \frac{27}{125} a - \frac{157}{375}$, $\frac{1}{40642491375} a^{14} - \frac{7}{40642491375} a^{13} - \frac{46003537}{40642491375} a^{12} - \frac{265878572}{40642491375} a^{11} + \frac{135659366}{40642491375} a^{10} + \frac{42906944}{40642491375} a^{9} + \frac{1266383537}{40642491375} a^{8} - \frac{3912440738}{40642491375} a^{7} - \frac{9052004897}{40642491375} a^{6} + \frac{14854406606}{40642491375} a^{5} - \frac{811648312}{8128498275} a^{4} - \frac{287758498}{8128498275} a^{3} + \frac{3907939841}{8128498275} a^{2} - \frac{1350597193}{40642491375} a + \frac{526715512}{8128498275}$, $\frac{1}{459869789908125} a^{15} + \frac{226}{18394791596325} a^{14} - \frac{10889472071}{153289929969375} a^{13} + \frac{565239157382}{459869789908125} a^{12} - \frac{1436201936813}{459869789908125} a^{11} + \frac{227023183297}{51096643323125} a^{10} + \frac{16937821594}{770301155625} a^{9} + \frac{1756518714782}{459869789908125} a^{8} - \frac{2273080130793}{51096643323125} a^{7} + \frac{170772239959771}{459869789908125} a^{6} - \frac{1354967851903}{459869789908125} a^{5} - \frac{18505450009471}{51096643323125} a^{4} + \frac{62385216074219}{153289929969375} a^{3} + \frac{41077902225364}{459869789908125} a^{2} - \frac{29402059155098}{153289929969375} a - \frac{5798307152132}{14834509351875}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( \frac{61}{11915125} a^{14} - \frac{427}{11915125} a^{13} + \frac{8128}{35745375} a^{12} - \frac{2141}{2383025} a^{11} + \frac{24032}{11915125} a^{10} - \frac{96623}{35745375} a^{9} + \frac{22701}{2383025} a^{8} - \frac{361083}{11915125} a^{7} + \frac{2230532}{35745375} a^{6} - \frac{1002577}{11915125} a^{5} - \frac{666604}{11915125} a^{4} + \frac{7793087}{35745375} a^{3} + \frac{3465329}{11915125} a^{2} - \frac{4873239}{11915125} a + \frac{22878742}{35745375} \) (order $6$)
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:  \( 137103.334761 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.SD_{16}$ (as 16T163):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $C_2^2.SD_{16}$
Character table for $C_2^2.SD_{16}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.549.1, 8.0.615710703429.2, 8.0.18385461.1, 8.0.10093618089.2

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$3.8.6.2$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$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$61$61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.8.6.1$x^{8} - 61 x^{4} + 59536$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$