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Properties

Label 16.0.15686332680...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 29^{4}\cdot 61^{4}$
Root discriminant $37.56$
Ramified primes $3, 5, 29, 61$
Class number $128$ (GRH)
Class group $[2, 4, 16]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T542)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![130321, 106495, 366837, 80477, 393392, -14352, 221779, -32084, 72905, -12451, 13752, -2204, 1357, -162, 62, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 62*x^14 - 162*x^13 + 1357*x^12 - 2204*x^11 + 13752*x^10 - 12451*x^9 + 72905*x^8 - 32084*x^7 + 221779*x^6 - 14352*x^5 + 393392*x^4 + 80477*x^3 + 366837*x^2 + 106495*x + 130321)
 
gp: K = bnfinit(x^16 - 4*x^15 + 62*x^14 - 162*x^13 + 1357*x^12 - 2204*x^11 + 13752*x^10 - 12451*x^9 + 72905*x^8 - 32084*x^7 + 221779*x^6 - 14352*x^5 + 393392*x^4 + 80477*x^3 + 366837*x^2 + 106495*x + 130321, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 62 x^{14} - 162 x^{13} + 1357 x^{12} - 2204 x^{11} + 13752 x^{10} - 12451 x^{9} + 72905 x^{8} - 32084 x^{7} + 221779 x^{6} - 14352 x^{5} + 393392 x^{4} + 80477 x^{3} + 366837 x^{2} + 106495 x + 130321 \)

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:  \(15686332680774922119140625=3^{8}\cdot 5^{12}\cdot 29^{4}\cdot 61^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.56$
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:  $3, 5, 29, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{19} a^{13} - \frac{1}{19} a^{12} + \frac{7}{19} a^{11} + \frac{6}{19} a^{10} + \frac{4}{19} a^{9} + \frac{4}{19} a^{8} + \frac{9}{19} a^{7} + \frac{3}{19} a^{6} - \frac{1}{19} a^{5} + \frac{6}{19} a^{3} - \frac{8}{19} a^{2} + \frac{3}{19} a$, $\frac{1}{209} a^{14} + \frac{4}{209} a^{13} - \frac{17}{209} a^{12} - \frac{16}{209} a^{11} + \frac{91}{209} a^{10} - \frac{52}{209} a^{9} + \frac{67}{209} a^{8} - \frac{104}{209} a^{7} + \frac{3}{19} a^{6} + \frac{52}{209} a^{5} + \frac{6}{209} a^{4} + \frac{98}{209} a^{3} - \frac{37}{209} a^{2} + \frac{91}{209} a - \frac{5}{11}$, $\frac{1}{8810933554239426661108012810319861} a^{15} + \frac{11844604916388394324111774535800}{8810933554239426661108012810319861} a^{14} + \frac{142781212672110297521414620483093}{8810933554239426661108012810319861} a^{13} + \frac{353895087686800350137253295242145}{8810933554239426661108012810319861} a^{12} - \frac{2753548056470446654638044270647805}{8810933554239426661108012810319861} a^{11} - \frac{46766779526949545714354603643754}{463733344959969824268842779490519} a^{10} - \frac{2116904120548896292442053969312046}{8810933554239426661108012810319861} a^{9} + \frac{1229577252638611750992766405363676}{8810933554239426661108012810319861} a^{8} + \frac{3578669524795750216164499566587582}{8810933554239426661108012810319861} a^{7} + \frac{2851009559040320864279898518642134}{8810933554239426661108012810319861} a^{6} - \frac{3560166065664609083292606926311843}{8810933554239426661108012810319861} a^{5} - \frac{4385030492830671986435756809948291}{8810933554239426661108012810319861} a^{4} + \frac{1363484540190004383473960293998406}{8810933554239426661108012810319861} a^{3} - \frac{343841598953612507636582427658688}{800993959476311514646182982756351} a^{2} + \frac{610916448042531958139933492183436}{8810933554239426661108012810319861} a - \frac{2350959415853102278087057032505}{24407018155787885487833830499501}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{16}$, which has order $128$ (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:  \( 7370.24268506 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T542):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 4.0.1990125.2, 4.0.68625.1, 8.4.32063125.1, 8.4.64927828125.4, 8.0.3960597515625.1

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$61$61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$