↑

Properties

Label 14.0.27627455695...0911.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,431^{7}$
Root discriminant $20.76$
Ramified prime $431$
Class number $3$
Class group $[3]$
Galois group $D_{7}$ (as 14T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![431, 0, 895, 0, 575, 0, 430, 0, 50, 0, 65, 0, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 6*x^12 + 65*x^10 + 50*x^8 + 430*x^6 + 575*x^4 + 895*x^2 + 431)
 
gp: K = bnfinit(x^14 + 6*x^12 + 65*x^10 + 50*x^8 + 430*x^6 + 575*x^4 + 895*x^2 + 431, 1)
 

Normalized defining polynomial

\( x^{14} + 6 x^{12} + 65 x^{10} + 50 x^{8} + 430 x^{6} + 575 x^{4} + 895 x^{2} + 431 \)

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:  \(-2762745569510280911=-\,431^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.76$
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:  $431$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{11} a^{6} + \frac{1}{11} a^{4} - \frac{4}{11} a^{2} + \frac{2}{11}$, $\frac{1}{22} a^{7} + \frac{1}{22} a^{5} - \frac{2}{11} a^{3} - \frac{1}{2} a^{2} - \frac{9}{22} a - \frac{1}{2}$, $\frac{1}{22} a^{8} - \frac{1}{22} a^{6} - \frac{3}{11} a^{4} - \frac{1}{2} a^{3} - \frac{1}{22} a^{2} - \frac{1}{2} a - \frac{2}{11}$, $\frac{1}{22} a^{9} - \frac{5}{22} a^{5} - \frac{1}{2} a^{4} - \frac{5}{22} a^{3} + \frac{9}{22} a - \frac{1}{2}$, $\frac{1}{242} a^{10} + \frac{3}{242} a^{8} - \frac{2}{121} a^{6} - \frac{1}{2} a^{5} - \frac{85}{242} a^{4} - \frac{1}{2} a^{3} + \frac{50}{121} a^{2} - \frac{46}{121}$, $\frac{1}{242} a^{11} + \frac{3}{242} a^{9} - \frac{2}{121} a^{7} - \frac{1}{22} a^{6} - \frac{85}{242} a^{5} - \frac{1}{22} a^{4} + \frac{50}{121} a^{3} + \frac{2}{11} a^{2} - \frac{46}{121} a - \frac{1}{11}$, $\frac{1}{939686} a^{12} - \frac{688}{469843} a^{10} - \frac{10194}{469843} a^{8} - \frac{7157}{469843} a^{6} - \frac{1}{2} a^{5} - \frac{14963}{85426} a^{4} - \frac{74738}{469843} a^{2} - \frac{239619}{939686}$, $\frac{1}{939686} a^{13} - \frac{688}{469843} a^{11} - \frac{10194}{469843} a^{9} - \frac{7157}{469843} a^{7} - \frac{1}{22} a^{6} - \frac{14963}{85426} a^{5} + \frac{5}{11} a^{4} - \frac{74738}{469843} a^{3} + \frac{2}{11} a^{2} - \frac{239619}{939686} a - \frac{1}{11}$

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:  \( 1508.64531505 \)
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

\(\Q(\sqrt{-431}) \), 7.1.80062991.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.80062991.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
431Data not computed