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Properties

Label 14.0.73107511205...2387.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,13^{7}\cdot 71^{13}$
Root discriminant $188.80$
Ramified primes $13, 71$
Class number $23110$ (GRH)
Class group $[23110]$ (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![215658439, 112363510, 226380237, 58323776, 52523677, 8576249, 5954484, 356239, 373778, 3351, 14741, -224, 216, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 216*x^12 - 224*x^11 + 14741*x^10 + 3351*x^9 + 373778*x^8 + 356239*x^7 + 5954484*x^6 + 8576249*x^5 + 52523677*x^4 + 58323776*x^3 + 226380237*x^2 + 112363510*x + 215658439)
 
gp: K = bnfinit(x^14 - x^13 + 216*x^12 - 224*x^11 + 14741*x^10 + 3351*x^9 + 373778*x^8 + 356239*x^7 + 5954484*x^6 + 8576249*x^5 + 52523677*x^4 + 58323776*x^3 + 226380237*x^2 + 112363510*x + 215658439, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 216 x^{12} - 224 x^{11} + 14741 x^{10} + 3351 x^{9} + 373778 x^{8} + 356239 x^{7} + 5954484 x^{6} + 8576249 x^{5} + 52523677 x^{4} + 58323776 x^{3} + 226380237 x^{2} + 112363510 x + 215658439 \)

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:  \(-73107511205515163512900457612387=-\,13^{7}\cdot 71^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $188.80$
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:  $13, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(923=13\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{923}(1,·)$, $\chi_{923}(261,·)$, $\chi_{923}(742,·)$, $\chi_{923}(456,·)$, $\chi_{923}(389,·)$, $\chi_{923}(534,·)$, $\chi_{923}(872,·)$, $\chi_{923}(467,·)$, $\chi_{923}(181,·)$, $\chi_{923}(662,·)$, $\chi_{923}(755,·)$, $\chi_{923}(922,·)$, $\chi_{923}(168,·)$, $\chi_{923}(51,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5472122338775984396851454651047894578593494469789240855} a^{13} - \frac{529363255411542860461836672462662046198134364082055977}{5472122338775984396851454651047894578593494469789240855} a^{12} + \frac{419121396716635545663297454140519491582384680108313222}{5472122338775984396851454651047894578593494469789240855} a^{11} + \frac{210436966143270806952817648909931203261398118656008211}{5472122338775984396851454651047894578593494469789240855} a^{10} + \frac{1305958275054464930347864274702823454741521475309698843}{5472122338775984396851454651047894578593494469789240855} a^{9} - \frac{488857359756610202033618823184547718582362405261985658}{5472122338775984396851454651047894578593494469789240855} a^{8} - \frac{549898437230596725517977991850925420469624264259738187}{5472122338775984396851454651047894578593494469789240855} a^{7} + \frac{1645644495066781387748023608917582197572083365572675784}{5472122338775984396851454651047894578593494469789240855} a^{6} - \frac{95058681912663032237847937566786996396480941330271409}{321889549339763788050085567708699681093734968811131815} a^{5} - \frac{1220113476563764487942423490383353177001481565250689647}{5472122338775984396851454651047894578593494469789240855} a^{4} + \frac{2694866653949071393093145709640673090544234291158754152}{5472122338775984396851454651047894578593494469789240855} a^{3} + \frac{1361946923300982927505122723708467694544576985693969116}{5472122338775984396851454651047894578593494469789240855} a^{2} + \frac{2256751640115259000723987060376646076930978023728867139}{5472122338775984396851454651047894578593494469789240855} a + \frac{348844599152602658756407464734463310570762609818771428}{1094424467755196879370290930209578915718698893957848171}$

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

Class group and class number

$C_{23110}$, which has order $23110$ (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:  \( 315114.6966253571 \) (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

\(\Q(\sqrt{-923}) \), 7.7.128100283921.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$13$13.14.7.2$x^{14} - 48268090 x^{2} + 125497034$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$71$71.14.13.1$x^{14} - 71$$14$$1$$13$$C_{14}$$[\ ]_{14}$