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Properties

Label 14.0.446035689155158947.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{9}\cdot 13^{2}\cdot 366181^{2}$
Root discriminant $18.23$
Ramified primes $3, 13, 366181$
Class number $1$
Class group Trivial
Galois group 14T52

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 45, 114, 36, 163, 111, 151, 70, 58, -5, -8, -14, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^12 - 14*x^11 - 8*x^10 - 5*x^9 + 58*x^8 + 70*x^7 + 151*x^6 + 111*x^5 + 163*x^4 + 36*x^3 + 114*x^2 + 45*x + 9)
 
gp: K = bnfinit(x^14 - 2*x^12 - 14*x^11 - 8*x^10 - 5*x^9 + 58*x^8 + 70*x^7 + 151*x^6 + 111*x^5 + 163*x^4 + 36*x^3 + 114*x^2 + 45*x + 9, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{12} - 14 x^{11} - 8 x^{10} - 5 x^{9} + 58 x^{8} + 70 x^{7} + 151 x^{6} + 111 x^{5} + 163 x^{4} + 36 x^{3} + 114 x^{2} + 45 x + 9 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-446035689155158947=-\,3^{9}\cdot 13^{2}\cdot 366181^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.23$
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:  $3, 13, 366181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{1}{9} a^{4}$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{5}$, $\frac{1}{27} a^{11} + \frac{1}{27} a^{9} + \frac{2}{27} a^{8} + \frac{4}{27} a^{7} - \frac{1}{27} a^{6} + \frac{1}{9} a^{5} - \frac{4}{27} a^{4} + \frac{1}{3} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{243} a^{12} - \frac{4}{243} a^{11} + \frac{10}{243} a^{10} - \frac{11}{243} a^{9} + \frac{23}{243} a^{8} + \frac{28}{243} a^{7} - \frac{38}{243} a^{6} - \frac{25}{243} a^{5} - \frac{83}{243} a^{4} - \frac{26}{81} a^{3} - \frac{19}{81} a^{2} + \frac{1}{27} a + \frac{7}{27}$, $\frac{1}{44469} a^{13} - \frac{32}{44469} a^{12} + \frac{473}{44469} a^{11} + \frac{623}{14823} a^{10} - \frac{1073}{44469} a^{9} + \frac{5783}{44469} a^{8} + \frac{1652}{14823} a^{7} + \frac{3982}{44469} a^{6} + \frac{2291}{44469} a^{5} - \frac{10066}{44469} a^{4} - \frac{2558}{14823} a^{3} - \frac{2678}{14823} a^{2} + \frac{803}{1647} a + \frac{1424}{4941}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $6$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1834}{44469} a^{13} + \frac{311}{44469} a^{12} + \frac{3415}{44469} a^{11} + \frac{2809}{4941} a^{10} + \frac{11063}{44469} a^{9} + \frac{8527}{44469} a^{8} - \frac{4096}{1647} a^{7} - \frac{114016}{44469} a^{6} - \frac{261887}{44469} a^{5} - \frac{148592}{44469} a^{4} - \frac{80719}{14823} a^{3} + \frac{2477}{14823} a^{2} - \frac{2189}{549} a - \frac{1304}{4941} \) (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:  \( 9313.85318657 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T52:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 56448
The 27 conjugacy class representatives for [L(7)^2]2=L(7)wr2
Character table for [L(7)^2]2=L(7)wr2 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \)

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

Sibling fields

Degree 16 sibling: data not computed
Degree 28 siblings: data not computed
Degree 42 sibling: data not computed
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.7.0.1$x^{7} - 10 x + 2$$1$$7$$0$$C_7$$[\ ]^{7}$
366181Data not computed