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Properties

Label 14.0.68274775432...9375.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 5^{7}\cdot 43^{12}$
Root discriminant $97.31$
Ramified primes $3, 5, 43$
Class number $1766$ (GRH)
Class group $[1766]$ (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![1228891, 602483, 589214, 99663, 123285, 20590, 22358, -399, 1576, -82, 290, -15, 1, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + x^12 - 15*x^11 + 290*x^10 - 82*x^9 + 1576*x^8 - 399*x^7 + 22358*x^6 + 20590*x^5 + 123285*x^4 + 99663*x^3 + 589214*x^2 + 602483*x + 1228891)
 
gp: K = bnfinit(x^14 - 5*x^13 + x^12 - 15*x^11 + 290*x^10 - 82*x^9 + 1576*x^8 - 399*x^7 + 22358*x^6 + 20590*x^5 + 123285*x^4 + 99663*x^3 + 589214*x^2 + 602483*x + 1228891, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} + x^{12} - 15 x^{11} + 290 x^{10} - 82 x^{9} + 1576 x^{8} - 399 x^{7} + 22358 x^{6} + 20590 x^{5} + 123285 x^{4} + 99663 x^{3} + 589214 x^{2} + 602483 x + 1228891 \)

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:  \(-6827477543251035514764609375=-\,3^{7}\cdot 5^{7}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.31$
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:  $3, 5, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(645=3\cdot 5\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{645}(256,·)$, $\chi_{645}(1,·)$, $\chi_{645}(226,·)$, $\chi_{645}(451,·)$, $\chi_{645}(164,·)$, $\chi_{645}(391,·)$, $\chi_{645}(299,·)$, $\chi_{645}(44,·)$, $\chi_{645}(269,·)$, $\chi_{645}(494,·)$, $\chi_{645}(16,·)$, $\chi_{645}(434,·)$, $\chi_{645}(121,·)$, $\chi_{645}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{9} - \frac{2}{7} a^{8} + \frac{1}{7} a^{7} - \frac{2}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{4} + \frac{2}{7} a^{3} - \frac{1}{7} a^{2} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{11} - \frac{3}{7} a^{9} - \frac{1}{7} a^{8} - \frac{1}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{9} - \frac{2}{7} a^{7} + \frac{2}{7} a^{6} - \frac{3}{7} a^{5} - \frac{1}{7} a^{4} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2} + \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{1036587382219329365830852194013} a^{13} + \frac{5556319178638077611066495699}{1036587382219329365830852194013} a^{12} - \frac{52635189871986457185441367}{1036587382219329365830852194013} a^{11} - \frac{35536934900243704265759893482}{1036587382219329365830852194013} a^{10} + \frac{285892500276526404232190207696}{1036587382219329365830852194013} a^{9} - \frac{370828557070706359706325012635}{1036587382219329365830852194013} a^{8} - \frac{298627483266556749588274903837}{1036587382219329365830852194013} a^{7} + \frac{454961177714920707098457920054}{1036587382219329365830852194013} a^{6} - \frac{319139008586708641367886436702}{1036587382219329365830852194013} a^{5} + \frac{36847606517324435821933301100}{148083911745618480832978884859} a^{4} + \frac{233254926197109767510070511485}{1036587382219329365830852194013} a^{3} - \frac{310841601602039708055303813573}{1036587382219329365830852194013} a^{2} + \frac{52063724705790597249500415118}{1036587382219329365830852194013} a - \frac{134307264869194666624686918487}{1036587382219329365830852194013}$

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

Class group and class number

$C_{1766}$, which has order $1766$ (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:  \( 35991.64185055774 \) (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

\(\Q(\sqrt{-15}) \), 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$3$3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$5$5.14.7.2$x^{14} - 15625 x^{2} + 156250$$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}$