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Properties

Label 14.2.346634844681273344.1
Degree $14$
Signature $[2, 6]$
Discriminant $2^{33}\cdot 7^{9}$
Root discriminant $17.90$
Ramified primes $2, 7$
Class number $1$
Class group Trivial
Galois group $D_7 \wr C_2$ (as 14T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, 0, -24, 104, -103, 94, -31, 64, -14, 12, 18, -8, 9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 9*x^12 - 8*x^11 + 18*x^10 + 12*x^9 - 14*x^8 + 64*x^7 - 31*x^6 + 94*x^5 - 103*x^4 + 104*x^3 - 24*x^2 + 8)
 
gp: K = bnfinit(x^14 - 2*x^13 + 9*x^12 - 8*x^11 + 18*x^10 + 12*x^9 - 14*x^8 + 64*x^7 - 31*x^6 + 94*x^5 - 103*x^4 + 104*x^3 - 24*x^2 + 8, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 9 x^{12} - 8 x^{11} + 18 x^{10} + 12 x^{9} - 14 x^{8} + 64 x^{7} - 31 x^{6} + 94 x^{5} - 103 x^{4} + 104 x^{3} - 24 x^{2} + 8 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(346634844681273344=2^{33}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.90$
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:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{28} a^{11} - \frac{3}{28} a^{10} + \frac{1}{28} a^{8} - \frac{1}{4} a^{6} + \frac{2}{7} a^{5} - \frac{1}{4} a^{4} - \frac{1}{28} a^{3} + \frac{3}{7} a^{2} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{56} a^{12} - \frac{1}{28} a^{10} - \frac{3}{28} a^{9} - \frac{1}{14} a^{8} - \frac{3}{28} a^{6} - \frac{9}{28} a^{5} + \frac{27}{56} a^{4} - \frac{3}{14} a^{3} - \frac{1}{14} a^{2} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{2604560} a^{13} - \frac{6353}{2604560} a^{12} - \frac{1423}{162785} a^{11} + \frac{5813}{65114} a^{10} + \frac{72019}{1302280} a^{9} + \frac{4741}{186040} a^{8} - \frac{16917}{651140} a^{7} - \frac{80501}{325570} a^{6} + \frac{284437}{2604560} a^{5} + \frac{640247}{2604560} a^{4} - \frac{57089}{130228} a^{3} + \frac{50934}{162785} a^{2} - \frac{102591}{325570} a + \frac{19483}{46510}$

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:  $7$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 13565.9141384 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7\wr C_2$ (as 14T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 392
The 20 conjugacy class representatives for $D_7 \wr C_2$
Character table for $D_7 \wr C_2$

Intermediate fields

\(\Q(\sqrt{2}) \)

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

Sibling fields

Degree 14 sibling: data not computed
Degree 28 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 ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$2$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.10.3$x^{4} + 6 x^{2} - 9$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.4.10.3$x^{4} + 6 x^{2} - 9$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.4.10.3$x^{4} + 6 x^{2} - 9$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
$7$7.7.9.1$x^{7} + 14 x^{3} + 7$$7$$1$$9$$D_{7}$$[3/2]_{2}$
7.7.0.1$x^{7} - x + 2$$1$$7$$0$$C_7$$[\ ]^{7}$