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Properties

Label 14.0.13423938470...9375.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,5^{7}\cdot 43^{13}$
Root discriminant $73.50$
Ramified primes $5, 43$
Class number $2842$ (GRH)
Class group $[2842]$ (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![119189, 169588, 270017, 181931, 129077, 71331, 36289, 17291, 4916, 2263, 585, 67, 45, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 45*x^12 + 67*x^11 + 585*x^10 + 2263*x^9 + 4916*x^8 + 17291*x^7 + 36289*x^6 + 71331*x^5 + 129077*x^4 + 181931*x^3 + 270017*x^2 + 169588*x + 119189)
 
gp: K = bnfinit(x^14 - x^13 + 45*x^12 + 67*x^11 + 585*x^10 + 2263*x^9 + 4916*x^8 + 17291*x^7 + 36289*x^6 + 71331*x^5 + 129077*x^4 + 181931*x^3 + 270017*x^2 + 169588*x + 119189, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 45 x^{12} + 67 x^{11} + 585 x^{10} + 2263 x^{9} + 4916 x^{8} + 17291 x^{7} + 36289 x^{6} + 71331 x^{5} + 129077 x^{4} + 181931 x^{3} + 270017 x^{2} + 169588 x + 119189 \)

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:  \(-134239384709553967597109375=-\,5^{7}\cdot 43^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.50$
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$$\chi_{215}(1,·)$, $\chi_{215}(194,·)$, $\chi_{215}(39,·)$, $\chi_{215}(41,·)$, $\chi_{215}(11,·)$, $\chi_{215}(204,·)$, $\chi_{215}(174,·)$, $\chi_{215}(16,·)$, $\chi_{215}(21,·)$, $\chi_{215}(214,·)$, $\chi_{215}(176,·)$, $\chi_{215}(121,·)$, $\chi_{215}(94,·)$, $\chi_{215}(199,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{49} a^{11} - \frac{1}{49} a^{10} - \frac{1}{49} a^{9} + \frac{3}{49} a^{8} - \frac{1}{49} a^{7} + \frac{2}{49} a^{6} + \frac{5}{49} a^{5} - \frac{9}{49} a^{4} - \frac{15}{49} a^{3} - \frac{9}{49} a^{2} + \frac{11}{49} a + \frac{2}{7}$, $\frac{1}{432115859} a^{12} - \frac{52848}{8818691} a^{11} + \frac{28721810}{432115859} a^{10} + \frac{604416}{11678807} a^{9} - \frac{30570188}{432115859} a^{8} + \frac{1270235}{432115859} a^{7} + \frac{3289570}{61730837} a^{6} + \frac{22530602}{432115859} a^{5} - \frac{177760733}{432115859} a^{4} + \frac{100351283}{432115859} a^{3} + \frac{55844280}{432115859} a^{2} + \frac{15814285}{432115859} a - \frac{12152817}{61730837}$, $\frac{1}{2263801075822778737} a^{13} + \frac{2052718092}{2263801075822778737} a^{12} - \frac{2424315223633767}{2263801075822778737} a^{11} + \frac{44511743041277427}{2263801075822778737} a^{10} + \frac{90781117925049802}{2263801075822778737} a^{9} + \frac{2042997947005605}{61183812860075101} a^{8} - \frac{137547498484783949}{2263801075822778737} a^{7} - \frac{100793863084943550}{2263801075822778737} a^{6} - \frac{1121110692844993881}{2263801075822778737} a^{5} + \frac{870637730832056561}{2263801075822778737} a^{4} + \frac{1089308918085472547}{2263801075822778737} a^{3} + \frac{10034905503186705}{46200021955566913} a^{2} - \frac{942190709284734827}{2263801075822778737} a + \frac{30329494762905306}{323400153688968391}$

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

Class group and class number

$C_{2842}$, which has order $2842$ (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:  \( 35991.6418506 \) (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

\(\Q(\sqrt{-215}) \), 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/7.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ 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$5.14.7.2$x^{14} - 15625 x^{2} + 156250$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$43$43.14.13.1$x^{14} - 43$$14$$1$$13$$C_{14}$$[\ ]_{14}$