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Properties

Label 14.14.3121846156...8125.1
Degree $14$
Signature $[14, 0]$
Discriminant $5^{7}\cdot 43^{12}$
Root discriminant $56.18$
Ramified primes $5, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -62, 529, 698, -7200, -3280, 9113, 3921, -3159, -1207, 470, 135, -34, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 - 34*x^12 + 135*x^11 + 470*x^10 - 1207*x^9 - 3159*x^8 + 3921*x^7 + 9113*x^6 - 3280*x^5 - 7200*x^4 + 698*x^3 + 529*x^2 - 62*x + 1)
 
gp: K = bnfinit(x^14 - 5*x^13 - 34*x^12 + 135*x^11 + 470*x^10 - 1207*x^9 - 3159*x^8 + 3921*x^7 + 9113*x^6 - 3280*x^5 - 7200*x^4 + 698*x^3 + 529*x^2 - 62*x + 1, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} - 34 x^{12} + 135 x^{11} + 470 x^{10} - 1207 x^{9} - 3159 x^{8} + 3921 x^{7} + 9113 x^{6} - 3280 x^{5} - 7200 x^{4} + 698 x^{3} + 529 x^{2} - 62 x + 1 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3121846156036138781328125=5^{7}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.18$
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:  $5, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(215=5\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{215}(64,·)$, $\chi_{215}(1,·)$, $\chi_{215}(164,·)$, $\chi_{215}(4,·)$, $\chi_{215}(41,·)$, $\chi_{215}(11,·)$, $\chi_{215}(44,·)$, $\chi_{215}(16,·)$, $\chi_{215}(84,·)$, $\chi_{215}(21,·)$, $\chi_{215}(54,·)$, $\chi_{215}(176,·)$, $\chi_{215}(121,·)$, $\chi_{215}(59,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{6} - \frac{2}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{11} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{4} - \frac{2}{7} a^{2} + \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{9} + \frac{1}{7} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{5} - \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{23182776913800847} a^{13} - \frac{41490120055044}{3311825273400121} a^{12} - \frac{782384907683440}{23182776913800847} a^{11} - \frac{328398103258082}{23182776913800847} a^{10} + \frac{5034581696623604}{23182776913800847} a^{9} - \frac{8751285786890532}{23182776913800847} a^{8} + \frac{9898043890659150}{23182776913800847} a^{7} - \frac{2463382391981797}{23182776913800847} a^{6} + \frac{9248663576302517}{23182776913800847} a^{5} + \frac{4136271079423410}{23182776913800847} a^{4} + \frac{7223191377097053}{23182776913800847} a^{3} + \frac{3176610345054164}{23182776913800847} a^{2} - \frac{1085390876010678}{3311825273400121} a - \frac{6919085416119146}{23182776913800847}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 32204571.9144 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{14}$ (as 14T1):

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

Intermediate fields

\(\Q(\sqrt{5}) \), 7.7.6321363049.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$5$5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$43$43.14.12.1$x^{14} + 3569 x^{7} + 4043763$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$