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Properties

Label 14.0.14498762356...2503.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{7}\cdot 127^{12}$
Root discriminant $168.19$
Ramified primes $7, 127$
Class number $8141$ (GRH)
Class group $[8141]$ (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![22851677, -14809311, -2368956, 5215705, 66341, -1228000, 150754, 150497, -23248, -12422, 2536, 413, -85, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 - 85*x^12 + 413*x^11 + 2536*x^10 - 12422*x^9 - 23248*x^8 + 150497*x^7 + 150754*x^6 - 1228000*x^5 + 66341*x^4 + 5215705*x^3 - 2368956*x^2 - 14809311*x + 22851677)
 
gp: K = bnfinit(x^14 - 5*x^13 - 85*x^12 + 413*x^11 + 2536*x^10 - 12422*x^9 - 23248*x^8 + 150497*x^7 + 150754*x^6 - 1228000*x^5 + 66341*x^4 + 5215705*x^3 - 2368956*x^2 - 14809311*x + 22851677, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} - 85 x^{12} + 413 x^{11} + 2536 x^{10} - 12422 x^{9} - 23248 x^{8} + 150497 x^{7} + 150754 x^{6} - 1228000 x^{5} + 66341 x^{4} + 5215705 x^{3} - 2368956 x^{2} - 14809311 x + 22851677 \)

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:  \(-14498762356636996870828289312503=-\,7^{7}\cdot 127^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $168.19$
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:  $7, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(889=7\cdot 127\)
Dirichlet character group:    $\lbrace$$\chi_{889}(512,·)$, $\chi_{889}(1,·)$, $\chi_{889}(258,·)$, $\chi_{889}(643,·)$, $\chi_{889}(8,·)$, $\chi_{889}(636,·)$, $\chi_{889}(778,·)$, $\chi_{889}(524,·)$, $\chi_{889}(64,·)$, $\chi_{889}(286,·)$, $\chi_{889}(540,·)$, $\chi_{889}(699,·)$, $\chi_{889}(764,·)$, $\chi_{889}(510,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19} a^{10} - \frac{9}{19} a^{9} + \frac{8}{19} a^{8} + \frac{9}{19} a^{7} - \frac{8}{19} a^{6} + \frac{3}{19} a^{5} - \frac{2}{19} a^{4} + \frac{1}{19} a^{3} + \frac{3}{19} a^{2} + \frac{6}{19} a + \frac{4}{19}$, $\frac{1}{703} a^{11} + \frac{13}{703} a^{10} - \frac{9}{37} a^{9} + \frac{204}{703} a^{8} - \frac{11}{37} a^{7} - \frac{78}{703} a^{6} + \frac{140}{703} a^{5} - \frac{43}{703} a^{4} + \frac{215}{703} a^{3} + \frac{167}{703} a^{2} - \frac{187}{703} a + \frac{107}{703}$, $\frac{1}{13357} a^{12} - \frac{7}{13357} a^{11} + \frac{235}{13357} a^{10} + \frac{4660}{13357} a^{9} - \frac{5991}{13357} a^{8} - \frac{1152}{13357} a^{7} - \frac{816}{13357} a^{6} - \frac{4360}{13357} a^{5} + \frac{2555}{13357} a^{4} - \frac{4170}{13357} a^{3} + \frac{580}{13357} a^{2} + \frac{3625}{13357} a + \frac{1930}{13357}$, $\frac{1}{3246617032231010311161396361543271} a^{13} + \frac{48311658449313957069857799553}{3246617032231010311161396361543271} a^{12} + \frac{683153096728992943587585879201}{3246617032231010311161396361543271} a^{11} + \frac{50103425423787702341356621756424}{3246617032231010311161396361543271} a^{10} + \frac{17924678957921191836949804883968}{87746406276513792193551253014683} a^{9} - \frac{165156025983411841266371423426437}{3246617032231010311161396361543271} a^{8} + \frac{101710668128469750162946067374996}{3246617032231010311161396361543271} a^{7} - \frac{1608319432300615792712222526190523}{3246617032231010311161396361543271} a^{6} + \frac{816122243927102458638447627680879}{3246617032231010311161396361543271} a^{5} - \frac{1605067972884087338424318088570488}{3246617032231010311161396361543271} a^{4} + \frac{1007783975907275222688470289333313}{3246617032231010311161396361543271} a^{3} + \frac{1032690364434278455171966621540104}{3246617032231010311161396361543271} a^{2} + \frac{1254894135767349006944601754408928}{3246617032231010311161396361543271} a + \frac{235167443990296240298850942516715}{3246617032231010311161396361543271}$

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

Class group and class number

$C_{8141}$, which has order $8141$ (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

\(\Q(\sqrt{-7}) \), 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$7$7.14.7.1$x^{14} - 117649 x^{2} + 1647086$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$127$127.7.6.1$x^{7} - 127$$7$$1$$6$$C_7$$[\ ]_{7}$
127.7.6.1$x^{7} - 127$$7$$1$$6$$C_7$$[\ ]_{7}$