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Properties

Label 16.0.17046596947...6416.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 3^{12}\cdot 23^{8}$
Root discriminant $18.39$
Ramified primes $2, 3, 23$
Class number $3$
Class group $[3]$
Galois group $C_8:C_2^2$ (as 16T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 14, 98, 337, 605, 556, 276, 150, 64, -84, -6, 4, -25, 4, 2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 2*x^14 + 4*x^13 - 25*x^12 + 4*x^11 - 6*x^10 - 84*x^9 + 64*x^8 + 150*x^7 + 276*x^6 + 556*x^5 + 605*x^4 + 337*x^3 + 98*x^2 + 14*x + 1)
 
gp: K = bnfinit(x^16 - x^15 + 2*x^14 + 4*x^13 - 25*x^12 + 4*x^11 - 6*x^10 - 84*x^9 + 64*x^8 + 150*x^7 + 276*x^6 + 556*x^5 + 605*x^4 + 337*x^3 + 98*x^2 + 14*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 2 x^{14} + 4 x^{13} - 25 x^{12} + 4 x^{11} - 6 x^{10} - 84 x^{9} + 64 x^{8} + 150 x^{7} + 276 x^{6} + 556 x^{5} + 605 x^{4} + 337 x^{3} + 98 x^{2} + 14 x + 1 \)

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:  \(170465969474436796416=2^{12}\cdot 3^{12}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.39$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{438} a^{13} + \frac{11}{146} a^{12} + \frac{15}{146} a^{11} + \frac{47}{219} a^{10} + \frac{97}{219} a^{9} - \frac{26}{73} a^{8} - \frac{13}{219} a^{7} + \frac{19}{219} a^{6} - \frac{13}{146} a^{5} - \frac{31}{219} a^{4} + \frac{70}{219} a^{3} + \frac{32}{73} a^{2} + \frac{45}{146} a - \frac{49}{438}$, $\frac{1}{438} a^{14} - \frac{11}{219} a^{12} + \frac{23}{146} a^{11} + \frac{85}{438} a^{10} + \frac{85}{438} a^{9} + \frac{2}{73} a^{8} + \frac{83}{219} a^{7} + \frac{47}{219} a^{6} + \frac{19}{146} a^{5} - \frac{77}{438} a^{4} + \frac{25}{438} a^{3} + \frac{77}{438} a^{2} + \frac{11}{219} a - \frac{104}{219}$, $\frac{1}{774231138} a^{15} + \frac{298268}{387115569} a^{14} - \frac{316441}{387115569} a^{13} + \frac{7961905}{258077046} a^{12} - \frac{79773883}{774231138} a^{11} - \frac{192811231}{774231138} a^{10} - \frac{8651677}{387115569} a^{9} + \frac{42958753}{387115569} a^{8} - \frac{1226456}{5302953} a^{7} + \frac{250324669}{774231138} a^{6} + \frac{111451919}{774231138} a^{5} + \frac{243892925}{774231138} a^{4} + \frac{32399169}{86025682} a^{3} + \frac{22593161}{387115569} a^{2} - \frac{88460758}{387115569} a + \frac{116312120}{387115569}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( -\frac{8524}{18177} a^{15} + \frac{32041}{36354} a^{14} - \frac{56209}{36354} a^{13} - \frac{28723}{36354} a^{12} + \frac{469363}{36354} a^{11} - \frac{233561}{18177} a^{10} + \frac{350407}{36354} a^{9} + \frac{625892}{18177} a^{8} - \frac{2294509}{36354} a^{7} - \frac{498856}{18177} a^{6} - \frac{3128831}{36354} a^{5} - \frac{3112799}{18177} a^{4} - \frac{1185733}{12118} a^{3} - \frac{34337}{12118} a^{2} + \frac{53113}{6059} a + \frac{61199}{36354} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6987.90665691 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T45):

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 $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{69}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-3}) \), 4.0.6348.1, 4.0.6348.2, \(\Q(\sqrt{-3}, \sqrt{-23})\), 8.0.362673936.1

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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$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}$
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$