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Properties

Label 14.0.10332321900...4416.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 349^{7}$
Root discriminant $37.36$
Ramified primes $2, 349$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_{7}$ (as 14T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9216, 3072, 4928, 13632, 10400, 2960, 2740, 2992, 1256, 244, 117, 16, -18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 18*x^12 + 16*x^11 + 117*x^10 + 244*x^9 + 1256*x^8 + 2992*x^7 + 2740*x^6 + 2960*x^5 + 10400*x^4 + 13632*x^3 + 4928*x^2 + 3072*x + 9216)
 
gp: K = bnfinit(x^14 - 4*x^13 - 18*x^12 + 16*x^11 + 117*x^10 + 244*x^9 + 1256*x^8 + 2992*x^7 + 2740*x^6 + 2960*x^5 + 10400*x^4 + 13632*x^3 + 4928*x^2 + 3072*x + 9216, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} - 18 x^{12} + 16 x^{11} + 117 x^{10} + 244 x^{9} + 1256 x^{8} + 2992 x^{7} + 2740 x^{6} + 2960 x^{5} + 10400 x^{4} + 13632 x^{3} + 4928 x^{2} + 3072 x + 9216 \)

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:  \(-10332321900000132284416=-\,2^{14}\cdot 349^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.36$
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:  $2, 349$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{24} a^{8} - \frac{1}{24} a^{7} + \frac{1}{24} a^{6} - \frac{1}{24} a^{5} - \frac{1}{12} a^{4} + \frac{1}{12} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{48} a^{9} - \frac{1}{16} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{3} a$, $\frac{1}{48} a^{10} - \frac{1}{16} a^{6} - \frac{1}{8} a^{4} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{192} a^{11} - \frac{1}{96} a^{9} - \frac{1}{64} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{11}{48} a^{3} - \frac{1}{2} a^{2} - \frac{1}{12} a$, $\frac{1}{1152} a^{12} - \frac{1}{576} a^{11} - \frac{1}{192} a^{10} - \frac{1}{288} a^{9} + \frac{13}{1152} a^{8} + \frac{7}{576} a^{7} + \frac{13}{288} a^{6} + \frac{5}{72} a^{5} + \frac{23}{96} a^{4} - \frac{31}{144} a^{3} + \frac{13}{72} a^{2} - \frac{1}{12} a$, $\frac{1}{686284900886016} a^{13} - \frac{33295985285}{343142450443008} a^{12} - \frac{856998917887}{343142450443008} a^{11} - \frac{215016354283}{171571225221504} a^{10} - \frac{1230018502705}{228761633628672} a^{9} - \frac{773938768405}{38126938938112} a^{8} - \frac{3285320994971}{57190408407168} a^{7} - \frac{2346540893999}{28595204203584} a^{6} + \frac{21274390721249}{171571225221504} a^{5} - \frac{17801438654881}{85785612610752} a^{4} + \frac{1208885331539}{42892806305376} a^{3} - \frac{8026455243769}{21446403152688} a^{2} + \frac{94061850679}{1787200262724} a + \frac{68356384088}{148933355227}$

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:  $6$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 9544377.94256 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7$ (as 14T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

\(\Q(\sqrt{-349}) \), 7.1.2720547136.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.2720547136.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$2$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
349Data not computed