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Properties

Label 14.0.17467807723...1875.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,5^{7}\cdot 127^{13}$
Root discriminant $200.92$
Ramified primes $5, 127$
Class number $31370$ (GRH)
Class group $[31370]$ (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![17221049, 33909513, 29806445, 12466019, 3273901, 1125862, 760329, 55438, 26433, 2085, 1915, 180, 132, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 132*x^12 + 180*x^11 + 1915*x^10 + 2085*x^9 + 26433*x^8 + 55438*x^7 + 760329*x^6 + 1125862*x^5 + 3273901*x^4 + 12466019*x^3 + 29806445*x^2 + 33909513*x + 17221049)
 
gp: K = bnfinit(x^14 - x^13 + 132*x^12 + 180*x^11 + 1915*x^10 + 2085*x^9 + 26433*x^8 + 55438*x^7 + 760329*x^6 + 1125862*x^5 + 3273901*x^4 + 12466019*x^3 + 29806445*x^2 + 33909513*x + 17221049, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 132 x^{12} + 180 x^{11} + 1915 x^{10} + 2085 x^{9} + 26433 x^{8} + 55438 x^{7} + 760329 x^{6} + 1125862 x^{5} + 3273901 x^{4} + 12466019 x^{3} + 29806445 x^{2} + 33909513 x + 17221049 \)

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:  \(-174678077231252895510919809921875=-\,5^{7}\cdot 127^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $200.92$
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:  $5, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(635=5\cdot 127\)
Dirichlet character group:    $\lbrace$$\chi_{635}(256,·)$, $\chi_{635}(1,·)$, $\chi_{635}(131,·)$, $\chi_{635}(516,·)$, $\chi_{635}(619,·)$, $\chi_{635}(16,·)$, $\chi_{635}(119,·)$, $\chi_{635}(504,·)$, $\chi_{635}(634,·)$, $\chi_{635}(379,·)$, $\chi_{635}(444,·)$, $\chi_{635}(349,·)$, $\chi_{635}(286,·)$, $\chi_{635}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{19} a^{8} + \frac{9}{19} a^{7} - \frac{8}{19} a^{6} + \frac{5}{19} a^{5} - \frac{8}{19} a^{4} + \frac{7}{19} a^{3} - \frac{9}{19} a^{2} + \frac{1}{19} a$, $\frac{1}{19} a^{9} + \frac{6}{19} a^{7} + \frac{1}{19} a^{6} + \frac{4}{19} a^{5} + \frac{3}{19} a^{4} + \frac{4}{19} a^{3} + \frac{6}{19} a^{2} - \frac{9}{19} a$, $\frac{1}{19} a^{10} + \frac{4}{19} a^{7} - \frac{5}{19} a^{6} - \frac{8}{19} a^{5} - \frac{5}{19} a^{4} + \frac{2}{19} a^{3} + \frac{7}{19} a^{2} - \frac{6}{19} a$, $\frac{1}{19} a^{11} - \frac{3}{19} a^{7} + \frac{5}{19} a^{6} - \frac{6}{19} a^{5} - \frac{4}{19} a^{4} - \frac{2}{19} a^{3} - \frac{8}{19} a^{2} - \frac{4}{19} a$, $\frac{1}{549803} a^{12} + \frac{3443}{549803} a^{11} + \frac{5884}{549803} a^{10} + \frac{13331}{549803} a^{9} - \frac{14139}{549803} a^{8} + \frac{102223}{549803} a^{7} + \frac{100319}{549803} a^{6} - \frac{249934}{549803} a^{5} + \frac{199339}{549803} a^{4} + \frac{28748}{549803} a^{3} + \frac{79186}{549803} a^{2} - \frac{161664}{549803} a - \frac{6692}{28937}$, $\frac{1}{275687328296854770669614616247135432657} a^{13} - \frac{114955326355545554699089877007823}{275687328296854770669614616247135432657} a^{12} + \frac{5994825703591747441239651972069756103}{275687328296854770669614616247135432657} a^{11} - \frac{3889253728322008109727884748548637985}{275687328296854770669614616247135432657} a^{10} + \frac{428136295980170037428391157249283600}{275687328296854770669614616247135432657} a^{9} - \frac{6745016312182377896227917032895659972}{275687328296854770669614616247135432657} a^{8} + \frac{1832974538357070051924227685456857401}{275687328296854770669614616247135432657} a^{7} + \frac{38250927821056761473894590755607101930}{275687328296854770669614616247135432657} a^{6} - \frac{27475376661811909472852500907413467861}{275687328296854770669614616247135432657} a^{5} - \frac{121619487055136809307793542744696539615}{275687328296854770669614616247135432657} a^{4} + \frac{33108946122071831950848083464210788585}{275687328296854770669614616247135432657} a^{3} - \frac{52026580196974519725542532237933487158}{275687328296854770669614616247135432657} a^{2} + \frac{84127003087138110524528188882009422696}{275687328296854770669614616247135432657} a + \frac{4994424999823464613249580801304903148}{14509859384044987929979716644586075403}$

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

Class group and class number

$C_{31370}$, which has order $31370$ (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:  \( 546287.2103473756 \) (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

\(\Q(\sqrt{-635}) \), 7.7.4195872914689.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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}$$[\ ]_{2}^{7}$
$127$127.14.13.1$x^{14} - 127$$14$$1$$13$$C_{14}$$[\ ]_{14}$