↑

Properties

Label 16.0.15544871745...4209.6
Degree $16$
Signature $[0, 8]$
Discriminant $47^{8}\cdot 97^{14}$
Root discriminant $375.38$
Ramified primes $47, 97$
Class number $46080$ (GRH)
Class group $[4, 16, 720]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8882874001, 0, -1443329186, 0, 1342090511, 0, -185395116, 0, 11771701, 0, -330492, 0, 7991, 0, -122, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 122*x^14 + 7991*x^12 - 330492*x^10 + 11771701*x^8 - 185395116*x^6 + 1342090511*x^4 - 1443329186*x^2 + 8882874001)
 
gp: K = bnfinit(x^16 - 122*x^14 + 7991*x^12 - 330492*x^10 + 11771701*x^8 - 185395116*x^6 + 1342090511*x^4 - 1443329186*x^2 + 8882874001, 1)
 

Normalized defining polynomial

\( x^{16} - 122 x^{14} + 7991 x^{12} - 330492 x^{10} + 11771701 x^{8} - 185395116 x^{6} + 1342090511 x^{4} - 1443329186 x^{2} + 8882874001 \)

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:  \(155448717458114694507131268341456449334209=47^{8}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $375.38$
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:  $47, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{12} a^{10} + \frac{1}{12} a^{8} - \frac{1}{12} a^{6} + \frac{1}{12} a^{4} + \frac{5}{12} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{12} a^{11} + \frac{1}{12} a^{9} - \frac{1}{12} a^{7} + \frac{1}{12} a^{5} + \frac{5}{12} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{792} a^{12} - \frac{1}{24} a^{11} - \frac{1}{33} a^{10} + \frac{1}{12} a^{9} - \frac{83}{792} a^{8} - \frac{1}{12} a^{7} - \frac{31}{792} a^{6} + \frac{1}{12} a^{5} + \frac{61}{792} a^{4} + \frac{5}{12} a^{3} + \frac{3}{11} a^{2} + \frac{7}{24} a - \frac{203}{792}$, $\frac{1}{243144} a^{13} + \frac{771}{27016} a^{11} + \frac{1237}{243144} a^{9} - \frac{1}{8} a^{8} - \frac{39763}{243144} a^{7} - \frac{1}{8} a^{6} - \frac{57623}{243144} a^{5} - \frac{1}{8} a^{4} - \frac{125}{20262} a^{3} - \frac{1}{8} a^{2} + \frac{14633}{121572} a + \frac{3}{8}$, $\frac{1}{1775997587556611521823883362904} a^{14} - \frac{233342790215436310652564143}{1775997587556611521823883362904} a^{12} + \frac{70861560214548456656183261785}{1775997587556611521823883362904} a^{10} - \frac{1}{8} a^{9} + \frac{7897797524260694081031802457}{161454326141510138347625760264} a^{8} - \frac{1}{8} a^{7} - \frac{57248555262234820422080335597}{1775997587556611521823883362904} a^{6} - \frac{1}{8} a^{5} - \frac{135063181606080131891464972259}{887998793778305760911941681452} a^{4} - \frac{1}{8} a^{3} - \frac{12799427646125331994078799035}{887998793778305760911941681452} a^{2} - \frac{1}{8} a + \frac{2635937463812156730953761}{9421837831470952062217548}$, $\frac{1}{545231259379879737199932192411528} a^{15} - \frac{233342790215436310652564143}{545231259379879737199932192411528} a^{13} + \frac{12354844874147778149271376521871}{545231259379879737199932192411528} a^{11} - \frac{1}{24} a^{10} - \frac{5373913073859410584173160206343}{49566478125443612472721108401048} a^{9} - \frac{1}{24} a^{8} + \frac{90222628812198850872291990612023}{545231259379879737199932192411528} a^{7} - \frac{5}{24} a^{6} + \frac{19104910683590544687867271459201}{272615629689939868599966096205764} a^{5} - \frac{1}{24} a^{4} + \frac{56227124178313239525762227692925}{272615629689939868599966096205764} a^{3} - \frac{5}{24} a^{2} + \frac{93895921239884335631766259}{723126053565395570775196809} a - \frac{1}{3}$

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

Class group and class number

$C_{4}\times C_{16}\times C_{720}$, which has order $46080$ (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:  \( 1817406770.17 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{97}) \), \(\Q(\sqrt{-47}) \), \(\Q(\sqrt{-4559}) \), 4.0.2016094657.2, 4.4.912673.1, \(\Q(\sqrt{-47}, \sqrt{97})\), 8.0.394269853600442921953.3 x2, 8.4.178483410412151617.2 x2, 8.0.4064637665983947649.3

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$47$47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$97$97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$