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Properties

Label 14.0.12228834564...7483.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,1987^{7}$
Root discriminant $44.58$
Ramified prime $1987$
Class number $169$
Class group $[13, 13]$
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![2012, -870, 7239, 5072, 8603, 7470, 4015, 2216, 1094, 30, 49, -88, 27, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 + 27*x^12 - 88*x^11 + 49*x^10 + 30*x^9 + 1094*x^8 + 2216*x^7 + 4015*x^6 + 7470*x^5 + 8603*x^4 + 5072*x^3 + 7239*x^2 - 870*x + 2012)
 
gp: K = bnfinit(x^14 - 6*x^13 + 27*x^12 - 88*x^11 + 49*x^10 + 30*x^9 + 1094*x^8 + 2216*x^7 + 4015*x^6 + 7470*x^5 + 8603*x^4 + 5072*x^3 + 7239*x^2 - 870*x + 2012, 1)
 

Normalized defining polynomial

\( x^{14} - 6 x^{13} + 27 x^{12} - 88 x^{11} + 49 x^{10} + 30 x^{9} + 1094 x^{8} + 2216 x^{7} + 4015 x^{6} + 7470 x^{5} + 8603 x^{4} + 5072 x^{3} + 7239 x^{2} - 870 x + 2012 \)

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:  \(-122288345645958900577483=-\,1987^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.58$
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:  $1987$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{12} a^{8} + \frac{1}{4} a^{2} - \frac{1}{3}$, $\frac{1}{24} a^{9} - \frac{1}{24} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{12} a + \frac{1}{6}$, $\frac{1}{96} a^{10} - \frac{1}{48} a^{9} - \frac{1}{32} a^{8} + \frac{1}{24} a^{7} + \frac{1}{48} a^{6} - \frac{1}{24} a^{5} + \frac{7}{32} a^{4} + \frac{1}{12} a^{3} + \frac{5}{32} a^{2} - \frac{1}{16} a - \frac{3}{8}$, $\frac{1}{192} a^{11} - \frac{1}{192} a^{10} + \frac{1}{64} a^{9} - \frac{7}{192} a^{8} + \frac{1}{32} a^{7} + \frac{7}{96} a^{6} - \frac{47}{192} a^{5} + \frac{13}{192} a^{4} - \frac{1}{192} a^{3} - \frac{79}{192} a^{2} - \frac{29}{96} a - \frac{3}{16}$, $\frac{1}{1152} a^{12} - \frac{1}{576} a^{11} - \frac{1}{288} a^{10} - \frac{5}{576} a^{9} - \frac{43}{1152} a^{8} + \frac{5}{144} a^{7} + \frac{83}{1152} a^{6} - \frac{1}{288} a^{5} - \frac{59}{576} a^{4} + \frac{17}{576} a^{3} - \frac{115}{1152} a^{2} - \frac{29}{576} a + \frac{49}{288}$, $\frac{1}{117072500128565760} a^{13} - \frac{7450141163189}{117072500128565760} a^{12} - \frac{105957020168993}{58536250064282880} a^{11} + \frac{43499070404639}{11707250012856576} a^{10} - \frac{1063368185679001}{117072500128565760} a^{9} - \frac{735356629178267}{117072500128565760} a^{8} + \frac{770845024188391}{23414500025713152} a^{7} - \frac{6595019556805669}{117072500128565760} a^{6} + \frac{5570631423456601}{58536250064282880} a^{5} - \frac{258347385663703}{3658515629017680} a^{4} - \frac{24160253532399349}{117072500128565760} a^{3} - \frac{19209246724614181}{117072500128565760} a^{2} + \frac{20859169991935181}{58536250064282880} a - \frac{751728213057}{6465236366720}$

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

Class group and class number

$C_{13}\times C_{13}$, which has order $169$

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:  \( 27094.4760334 \)
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

\(\Q(\sqrt{-1987}) \), 7.1.7845011803.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.7845011803.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
1987Data not computed