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Properties

Label 14.0.35651316359...9567.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,863^{7}$
Root discriminant $29.38$
Ramified prime $863$
Class number $3$
Class group $[3]$
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![26759, -35471, 36718, -32768, 20359, -11969, 5781, -2393, 1126, -467, 212, -70, 19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 + 19*x^12 - 70*x^11 + 212*x^10 - 467*x^9 + 1126*x^8 - 2393*x^7 + 5781*x^6 - 11969*x^5 + 20359*x^4 - 32768*x^3 + 36718*x^2 - 35471*x + 26759)
 
gp: K = bnfinit(x^14 - 6*x^13 + 19*x^12 - 70*x^11 + 212*x^10 - 467*x^9 + 1126*x^8 - 2393*x^7 + 5781*x^6 - 11969*x^5 + 20359*x^4 - 32768*x^3 + 36718*x^2 - 35471*x + 26759, 1)
 

Normalized defining polynomial

\( x^{14} - 6 x^{13} + 19 x^{12} - 70 x^{11} + 212 x^{10} - 467 x^{9} + 1126 x^{8} - 2393 x^{7} + 5781 x^{6} - 11969 x^{5} + 20359 x^{4} - 32768 x^{3} + 36718 x^{2} - 35471 x + 26759 \)

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:  \(-356513163590850929567=-\,863^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.38$
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:  $863$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{1685348665} a^{12} + \frac{1068941}{21887645} a^{11} + \frac{24401194}{1685348665} a^{10} - \frac{50376262}{1685348665} a^{9} + \frac{118730753}{240764095} a^{8} - \frac{90319233}{1685348665} a^{7} + \frac{606088178}{1685348665} a^{6} - \frac{462809374}{1685348665} a^{5} - \frac{303190803}{1685348665} a^{4} + \frac{11018454}{41106065} a^{3} - \frac{79901322}{1685348665} a^{2} - \frac{775881739}{1685348665} a + \frac{97784238}{337069733}$, $\frac{1}{24787904237631535} a^{13} + \frac{1033531}{3541129176804505} a^{12} + \frac{1984724692613249}{24787904237631535} a^{11} + \frac{14032774771202}{173341987675745} a^{10} + \frac{472324364511}{54478910412377} a^{9} - \frac{787202713927141}{2253445839784685} a^{8} - \frac{2910142838607344}{24787904237631535} a^{7} + \frac{3527419341783394}{24787904237631535} a^{6} + \frac{1627232037241}{10481143440859} a^{5} + \frac{3823738042861038}{24787904237631535} a^{4} + \frac{55391765518}{512962859045} a^{3} - \frac{7650347325477118}{24787904237631535} a^{2} - \frac{770244759613023}{1906761864433195} a + \frac{1522962173157872}{3541129176804505}$

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:  \( 11443.2128107 \)
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

\(\Q(\sqrt{-863}) \), 7.1.642735647.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.642735647.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
863Data not computed