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Properties

Label 14.0.14894671927...9267.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 281^{13}$
Root discriminant $325.36$
Ramified primes $3, 281$
Class number $190518$ (GRH)
Class group $[190518]$ (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![786888991, -423159590, 553225015, -228502327, 114730846, -21756177, 5480474, 230853, 7130, 40327, 2926, 840, 151, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 151*x^12 + 840*x^11 + 2926*x^10 + 40327*x^9 + 7130*x^8 + 230853*x^7 + 5480474*x^6 - 21756177*x^5 + 114730846*x^4 - 228502327*x^3 + 553225015*x^2 - 423159590*x + 786888991)
 
gp: K = bnfinit(x^14 - x^13 + 151*x^12 + 840*x^11 + 2926*x^10 + 40327*x^9 + 7130*x^8 + 230853*x^7 + 5480474*x^6 - 21756177*x^5 + 114730846*x^4 - 228502327*x^3 + 553225015*x^2 - 423159590*x + 786888991, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 151 x^{12} + 840 x^{11} + 2926 x^{10} + 40327 x^{9} + 7130 x^{8} + 230853 x^{7} + 5480474 x^{6} - 21756177 x^{5} + 114730846 x^{4} - 228502327 x^{3} + 553225015 x^{2} - 423159590 x + 786888991 \)

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:  \(-148946719271275003088728003994769267=-\,3^{7}\cdot 281^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $325.36$
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:  $3, 281$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(843=3\cdot 281\)
Dirichlet character group:    $\lbrace$$\chi_{843}(32,·)$, $\chi_{843}(1,·)$, $\chi_{843}(503,·)$, $\chi_{843}(842,·)$, $\chi_{843}(811,·)$, $\chi_{843}(109,·)$, $\chi_{843}(79,·)$, $\chi_{843}(116,·)$, $\chi_{843}(181,·)$, $\chi_{843}(662,·)$, $\chi_{843}(727,·)$, $\chi_{843}(340,·)$, $\chi_{843}(764,·)$, $\chi_{843}(734,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{49} a^{10} + \frac{2}{49} a^{9} + \frac{2}{49} a^{7} + \frac{1}{49} a^{6} + \frac{4}{49} a^{5} + \frac{8}{49} a^{4} + \frac{13}{49} a^{3} - \frac{10}{49} a^{2} - \frac{3}{7} a$, $\frac{1}{343} a^{11} + \frac{1}{343} a^{10} + \frac{19}{343} a^{9} - \frac{5}{343} a^{8} + \frac{13}{343} a^{7} - \frac{11}{343} a^{6} - \frac{3}{343} a^{5} + \frac{75}{343} a^{4} - \frac{58}{343} a^{3} + \frac{136}{343} a^{2} + \frac{4}{49} a + \frac{3}{7}$, $\frac{1}{343} a^{12} - \frac{3}{343} a^{10} - \frac{17}{343} a^{9} + \frac{18}{343} a^{8} - \frac{17}{343} a^{7} - \frac{13}{343} a^{6} - \frac{6}{343} a^{5} + \frac{6}{49} a^{4} - \frac{128}{343} a^{3} + \frac{102}{343} a^{2} + \frac{24}{49} a - \frac{3}{7}$, $\frac{1}{8498985255585801129845889728572566150706644767} a^{13} + \frac{1272199522992577046256185558759907324798056}{1214140750797971589977984246938938021529520681} a^{12} + \frac{1993378649895567088121432688636082446007907}{8498985255585801129845889728572566150706644767} a^{11} + \frac{71131573274404215679152020428365618444526226}{8498985255585801129845889728572566150706644767} a^{10} - \frac{485162680478966552140078891535631778155885097}{8498985255585801129845889728572566150706644767} a^{9} + \frac{382110169068870390346892652335298116401921152}{8498985255585801129845889728572566150706644767} a^{8} + \frac{540454495484524104432057075073403723771840959}{8498985255585801129845889728572566150706644767} a^{7} - \frac{107563783632624384939448722089834918172019887}{8498985255585801129845889728572566150706644767} a^{6} - \frac{35895873642399707987158006289375304842761892}{1214140750797971589977984246938938021529520681} a^{5} + \frac{2995925827319768177589972809703031856920912835}{8498985255585801129845889728572566150706644767} a^{4} - \frac{2502364972144825502618232917828430241047945224}{8498985255585801129845889728572566150706644767} a^{3} - \frac{334562013638626967213346100097838608604703868}{1214140750797971589977984246938938021529520681} a^{2} - \frac{55800386497784883725952270695724987597245961}{173448678685424512853997749562705431647074383} a + \frac{165873757382494707445421306978626399746650}{3539768952763765568448933664545008809123967}$

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

Class group and class number

$C_{190518}$, which has order $190518$ (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:  \( 12176100.831221418 \) (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

\(\Q(\sqrt{-843}) \), 7.7.492309163417681.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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}$
281Data not computed