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Properties

Label 14.0.25140949180...8391.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,631^{13}$
Root discriminant $398.13$
Ramified prime $631$
Class number $3108937$ (GRH)
Class group $[3108937]$ (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![20455038976, 4799748096, -2583896960, -414082048, 243943976, 26925836, -15658566, -1444591, 490605, 31737, -5095, -277, 23, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 23*x^12 - 277*x^11 - 5095*x^10 + 31737*x^9 + 490605*x^8 - 1444591*x^7 - 15658566*x^6 + 26925836*x^5 + 243943976*x^4 - 414082048*x^3 - 2583896960*x^2 + 4799748096*x + 20455038976)
 
gp: K = bnfinit(x^14 - x^13 + 23*x^12 - 277*x^11 - 5095*x^10 + 31737*x^9 + 490605*x^8 - 1444591*x^7 - 15658566*x^6 + 26925836*x^5 + 243943976*x^4 - 414082048*x^3 - 2583896960*x^2 + 4799748096*x + 20455038976, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 23 x^{12} - 277 x^{11} - 5095 x^{10} + 31737 x^{9} + 490605 x^{8} - 1444591 x^{7} - 15658566 x^{6} + 26925836 x^{5} + 243943976 x^{4} - 414082048 x^{3} - 2583896960 x^{2} + 4799748096 x + 20455038976 \)

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:  \(-2514094918068134291976760322363658391=-\,631^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $398.13$
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:  $631$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(631\)
Dirichlet character group:    $\lbrace$$\chi_{631}(1,·)$, $\chi_{631}(610,·)$, $\chi_{631}(133,·)$, $\chi_{631}(362,·)$, $\chi_{631}(427,·)$, $\chi_{631}(204,·)$, $\chi_{631}(269,·)$, $\chi_{631}(190,·)$, $\chi_{631}(498,·)$, $\chi_{631}(21,·)$, $\chi_{631}(630,·)$, $\chi_{631}(441,·)$, $\chi_{631}(601,·)$, $\chi_{631}(30,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{32} a^{7} - \frac{1}{32} a^{6} - \frac{1}{32} a^{5} + \frac{1}{32} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{128} a^{8} - \frac{1}{64} a^{6} + \frac{1}{32} a^{5} + \frac{5}{128} a^{4} - \frac{5}{32} a^{3} + \frac{7}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{256} a^{9} - \frac{1}{256} a^{8} - \frac{1}{128} a^{7} + \frac{3}{128} a^{6} + \frac{1}{256} a^{5} + \frac{7}{256} a^{4} - \frac{1}{16} a^{3} + \frac{13}{64} a^{2} + \frac{5}{16} a - \frac{1}{2}$, $\frac{1}{1024} a^{10} + \frac{1}{1024} a^{9} - \frac{1}{256} a^{8} + \frac{1}{512} a^{7} - \frac{19}{1024} a^{6} - \frac{55}{1024} a^{5} + \frac{15}{512} a^{4} + \frac{21}{256} a^{3} + \frac{23}{128} a^{2} + \frac{1}{32} a - \frac{1}{4}$, $\frac{1}{88064} a^{11} - \frac{13}{88064} a^{10} + \frac{23}{44032} a^{9} - \frac{139}{44032} a^{8} - \frac{303}{88064} a^{7} + \frac{2163}{88064} a^{6} + \frac{149}{2752} a^{5} - \frac{15}{2752} a^{4} + \frac{115}{2752} a^{3} + \frac{1029}{5504} a^{2} + \frac{81}{1376} a - \frac{41}{172}$, $\frac{1}{1937408} a^{12} + \frac{3}{968704} a^{11} + \frac{573}{1937408} a^{10} + \frac{685}{968704} a^{9} + \frac{6455}{1937408} a^{8} - \frac{93}{88064} a^{7} + \frac{22903}{1937408} a^{6} + \frac{10011}{968704} a^{5} - \frac{17915}{484352} a^{4} - \frac{60195}{242176} a^{3} - \frac{287}{7568} a^{2} + \frac{1831}{15136} a - \frac{153}{1892}$, $\frac{1}{6123513831409419146435989705427935232} a^{13} - \frac{124454882077837171050493447589}{3061756915704709573217994852713967616} a^{12} - \frac{8718210589619538333698340597625}{6123513831409419146435989705427935232} a^{11} - \frac{417156496869483883571647556875233}{1530878457852354786608997426356983808} a^{10} - \frac{9185633424947080590090237900843369}{6123513831409419146435989705427935232} a^{9} + \frac{682997554554130226613893187852899}{3061756915704709573217994852713967616} a^{8} - \frac{66570372879812017014232146740279031}{6123513831409419146435989705427935232} a^{7} - \frac{20340649685717436504813889464663747}{765439228926177393304498713178491904} a^{6} + \frac{5857032046593639206890238534281169}{765439228926177393304498713178491904} a^{5} - \frac{10646463018615804533457252783210855}{191359807231544348326124678294622976} a^{4} + \frac{6945716498885229009714026060032765}{382719614463088696652249356589245952} a^{3} - \frac{5422659242393065933047866012793481}{47839951807886087081531169573655744} a^{2} + \frac{8046467514383735268566203463152199}{23919975903943043540765584786827872} a + \frac{686473150752317730123885105434675}{2989996987992880442595698098353484}$

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

Class group and class number

$C_{3108937}$, which has order $3108937$ (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:  \( 10178543858.108408 \) (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

\(\Q(\sqrt{-631}) \), 7.7.63121332085847281.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
631Data not computed