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Properties

Label 14.0.82184209159918343.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,17^{6}\cdot 23^{7}$
Root discriminant $16.15$
Ramified primes $17, 23$
Class number $3$
Class group $[3]$
Galois group $D_{14}$ (as 14T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, 8, 62, 20, 78, -21, 61, -36, 47, -25, 23, -8, 7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 7*x^12 - 8*x^11 + 23*x^10 - 25*x^9 + 47*x^8 - 36*x^7 + 61*x^6 - 21*x^5 + 78*x^4 + 20*x^3 + 62*x^2 + 8*x + 27)
 
gp: K = bnfinit(x^14 - x^13 + 7*x^12 - 8*x^11 + 23*x^10 - 25*x^9 + 47*x^8 - 36*x^7 + 61*x^6 - 21*x^5 + 78*x^4 + 20*x^3 + 62*x^2 + 8*x + 27, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 7 x^{12} - 8 x^{11} + 23 x^{10} - 25 x^{9} + 47 x^{8} - 36 x^{7} + 61 x^{6} - 21 x^{5} + 78 x^{4} + 20 x^{3} + 62 x^{2} + 8 x + 27 \)

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:  \(-82184209159918343=-\,17^{6}\cdot 23^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.15$
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:  $17, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{2}{9} a^{5} + \frac{4}{9} a^{4} - \frac{2}{9} a^{3} - \frac{2}{9} a^{2} + \frac{1}{9} a$, $\frac{1}{621} a^{12} - \frac{5}{621} a^{11} + \frac{1}{27} a^{10} - \frac{80}{621} a^{9} + \frac{44}{621} a^{8} - \frac{88}{621} a^{7} - \frac{191}{621} a^{6} + \frac{5}{69} a^{5} - \frac{61}{207} a^{4} - \frac{10}{69} a^{3} - \frac{31}{69} a^{2} + \frac{47}{621} a + \frac{6}{23}$, $\frac{1}{80109} a^{13} - \frac{8}{26703} a^{12} + \frac{22}{1863} a^{11} + \frac{3002}{80109} a^{10} - \frac{292}{3483} a^{9} - \frac{3344}{26703} a^{8} + \frac{7484}{80109} a^{7} - \frac{19303}{80109} a^{6} - \frac{1243}{26703} a^{5} - \frac{1217}{26703} a^{4} + \frac{490}{2967} a^{3} + \frac{8039}{80109} a^{2} + \frac{27628}{80109} a + \frac{875}{2967}$

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:  $6$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 345.555600672 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{14}$ (as 14T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 28
The 10 conjugacy class representatives for $D_{14}$
Character table for $D_{14}$

Intermediate fields

\(\Q(\sqrt{-23}) \), 7.1.59776471.1

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 14 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_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}$