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Properties

Label 16.0.28320098225...1441.3
Degree $16$
Signature $[0, 8]$
Discriminant $7^{8}\cdot 53^{12}$
Root discriminant $51.97$
Ramified primes $7, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![231623, 76048, -157563, -47978, -16736, -22802, 14311, 1374, 2927, 6422, 679, -324, 216, 36, -25, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 25*x^14 + 36*x^13 + 216*x^12 - 324*x^11 + 679*x^10 + 6422*x^9 + 2927*x^8 + 1374*x^7 + 14311*x^6 - 22802*x^5 - 16736*x^4 - 47978*x^3 - 157563*x^2 + 76048*x + 231623)
 
gp: K = bnfinit(x^16 - 2*x^15 - 25*x^14 + 36*x^13 + 216*x^12 - 324*x^11 + 679*x^10 + 6422*x^9 + 2927*x^8 + 1374*x^7 + 14311*x^6 - 22802*x^5 - 16736*x^4 - 47978*x^3 - 157563*x^2 + 76048*x + 231623, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 25 x^{14} + 36 x^{13} + 216 x^{12} - 324 x^{11} + 679 x^{10} + 6422 x^{9} + 2927 x^{8} + 1374 x^{7} + 14311 x^{6} - 22802 x^{5} - 16736 x^{4} - 47978 x^{3} - 157563 x^{2} + 76048 x + 231623 \)

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:  \(2832009822518079193000591441=7^{8}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.97$
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:  $7, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{14} a^{8} + \frac{1}{14} a^{7} + \frac{2}{7} a^{6} + \frac{3}{14} a^{5} - \frac{5}{14} a^{4} - \frac{5}{14} a^{3} - \frac{3}{7} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{14} a^{9} + \frac{3}{14} a^{7} - \frac{1}{14} a^{6} + \frac{3}{7} a^{5} - \frac{1}{14} a^{3} - \frac{1}{14} a^{2} - \frac{1}{2}$, $\frac{1}{14} a^{10} - \frac{2}{7} a^{7} - \frac{3}{7} a^{6} + \frac{5}{14} a^{5} + \frac{2}{7} a^{2} - \frac{1}{2}$, $\frac{1}{14} a^{11} - \frac{1}{7} a^{7} - \frac{1}{2} a^{6} - \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{1}{2} a$, $\frac{1}{98} a^{12} - \frac{1}{49} a^{10} + \frac{1}{98} a^{9} - \frac{11}{49} a^{7} + \frac{45}{98} a^{6} + \frac{5}{49} a^{5} - \frac{13}{49} a^{4} - \frac{1}{2} a^{3} - \frac{3}{7} a^{2} - \frac{1}{2}$, $\frac{1}{98} a^{13} - \frac{1}{49} a^{11} + \frac{1}{98} a^{10} - \frac{1}{98} a^{8} - \frac{16}{49} a^{7} - \frac{2}{49} a^{6} + \frac{37}{98} a^{5} + \frac{3}{7} a^{4} - \frac{1}{2} a^{3} - \frac{2}{7} a^{2} - \frac{1}{2}$, $\frac{1}{1173746} a^{14} - \frac{445}{586873} a^{13} + \frac{807}{167678} a^{12} - \frac{14933}{586873} a^{11} - \frac{18451}{586873} a^{10} - \frac{16523}{586873} a^{9} + \frac{36173}{1173746} a^{8} + \frac{519321}{1173746} a^{7} - \frac{16641}{1173746} a^{6} - \frac{179231}{1173746} a^{5} + \frac{168541}{586873} a^{4} - \frac{26753}{83839} a^{3} + \frac{7664}{83839} a^{2} + \frac{437}{23954} a - \frac{143}{413}$, $\frac{1}{13284866524916417919642089278} a^{15} + \frac{1946562403000297876047}{13284866524916417919642089278} a^{14} - \frac{12199117280962596731382319}{6642433262458208959821044639} a^{13} - \frac{28875745357333756871228003}{6642433262458208959821044639} a^{12} - \frac{18919461823328275211141221}{6642433262458208959821044639} a^{11} + \frac{35366745710837966371745811}{1897838074988059702806012754} a^{10} + \frac{2352716592620011163335990}{948919037494029851403006377} a^{9} + \frac{5501762601849641911114874}{390731368379894644695355567} a^{8} - \frac{144730109960602178609958915}{13284866524916417919642089278} a^{7} - \frac{6527812385541425498007752161}{13284866524916417919642089278} a^{6} - \frac{5655221358766931865606830695}{13284866524916417919642089278} a^{5} + \frac{1332844313288912693434437543}{13284866524916417919642089278} a^{4} - \frac{608946532620164464393833077}{1897838074988059702806012754} a^{3} + \frac{852066513435374648974877329}{1897838074988059702806012754} a^{2} + \frac{20917424802668499693162202}{135559862499147121629000911} a - \frac{3276326327197579022088033}{9348956034423939422689718}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 21203688.514 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-371}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-7}, \sqrt{53})\), 4.2.19663.1 x2, 4.0.2597.2 x2, 8.0.18945044881.1, 8.2.7602375867247.2 x4

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
$53$53.4.3.1$x^{4} - 53$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.1$x^{4} - 53$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.1$x^{4} - 53$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.1$x^{4} - 53$$4$$1$$3$$C_4$$[\ ]_{4}$