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Properties

Label 16.0.12003260674...9641.8
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 61^{6}$
Root discriminant $31.99$
Ramified primes $13, 61$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4\wr C_2$ (as 16T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1381, 478, -1969, -191, 3937, -4041, 286, 1511, -1111, 617, -29, -148, 139, -70, 25, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 25*x^14 - 70*x^13 + 139*x^12 - 148*x^11 - 29*x^10 + 617*x^9 - 1111*x^8 + 1511*x^7 + 286*x^6 - 4041*x^5 + 3937*x^4 - 191*x^3 - 1969*x^2 + 478*x + 1381)
 
gp: K = bnfinit(x^16 - 7*x^15 + 25*x^14 - 70*x^13 + 139*x^12 - 148*x^11 - 29*x^10 + 617*x^9 - 1111*x^8 + 1511*x^7 + 286*x^6 - 4041*x^5 + 3937*x^4 - 191*x^3 - 1969*x^2 + 478*x + 1381, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 25 x^{14} - 70 x^{13} + 139 x^{12} - 148 x^{11} - 29 x^{10} + 617 x^{9} - 1111 x^{8} + 1511 x^{7} + 286 x^{6} - 4041 x^{5} + 3937 x^{4} - 191 x^{3} - 1969 x^{2} + 478 x + 1381 \)

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:  \(1200326067404665657109641=13^{12}\cdot 61^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.99$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} + \frac{6}{13} a^{11} + \frac{4}{13} a^{10} - \frac{1}{13} a^{9} - \frac{2}{13} a^{7} - \frac{2}{13} a^{6} - \frac{1}{13} a^{5} - \frac{3}{13} a^{4} + \frac{4}{13} a^{3} + \frac{4}{13} a^{2} - \frac{1}{13} a + \frac{1}{13}$, $\frac{1}{793} a^{13} + \frac{3}{793} a^{12} + \frac{103}{793} a^{11} + \frac{12}{61} a^{10} - \frac{166}{793} a^{9} + \frac{24}{793} a^{8} - \frac{217}{793} a^{7} - \frac{47}{793} a^{6} + \frac{11}{61} a^{5} + \frac{17}{61} a^{4} - \frac{8}{793} a^{3} + \frac{14}{61} a^{2} - \frac{334}{793} a + \frac{101}{793}$, $\frac{1}{793} a^{14} - \frac{28}{793} a^{12} - \frac{92}{793} a^{11} - \frac{329}{793} a^{10} - \frac{149}{793} a^{9} - \frac{289}{793} a^{8} + \frac{55}{793} a^{7} - \frac{265}{793} a^{6} - \frac{86}{793} a^{5} - \frac{5}{13} a^{4} - \frac{282}{793} a^{3} + \frac{218}{793} a^{2} - \frac{361}{793} a + \frac{368}{793}$, $\frac{1}{2250067043275045278323} a^{15} + \frac{948956490694852667}{2250067043275045278323} a^{14} - \frac{258967424654428426}{2250067043275045278323} a^{13} + \frac{78497755769086081032}{2250067043275045278323} a^{12} - \frac{325068700338520066997}{2250067043275045278323} a^{11} + \frac{1049282971339568899744}{2250067043275045278323} a^{10} + \frac{62933770761076555020}{173082080251926559871} a^{9} - \frac{1047655448660692999363}{2250067043275045278323} a^{8} - \frac{726134600355839363324}{2250067043275045278323} a^{7} - \frac{811429721031293456101}{2250067043275045278323} a^{6} - \frac{747449131964085153637}{2250067043275045278323} a^{5} + \frac{1066840827547778122128}{2250067043275045278323} a^{4} + \frac{895081169857045118501}{2250067043275045278323} a^{3} - \frac{1114307696432538779936}{2250067043275045278323} a^{2} + \frac{302580266603390893914}{2250067043275045278323} a + \frac{463631853263634214817}{2250067043275045278323}$

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:  \( 113920.750816 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T28):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 4.4.10309.1, 4.0.134017.1, 8.0.17960556289.2

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 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$