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Properties

Label 14.0.34413598978...1632.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 71^{12}$
Root discriminant $109.23$
Ramified primes $2, 71$
Class number $1247$ (GRH)
Class group $[1247]$ (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![1953011, 1336242, 987997, 411952, 206418, 151674, 43102, -20276, -11085, 632, 1030, 30, -45, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 45*x^12 + 30*x^11 + 1030*x^10 + 632*x^9 - 11085*x^8 - 20276*x^7 + 43102*x^6 + 151674*x^5 + 206418*x^4 + 411952*x^3 + 987997*x^2 + 1336242*x + 1953011)
 
gp: K = bnfinit(x^14 - 2*x^13 - 45*x^12 + 30*x^11 + 1030*x^10 + 632*x^9 - 11085*x^8 - 20276*x^7 + 43102*x^6 + 151674*x^5 + 206418*x^4 + 411952*x^3 + 987997*x^2 + 1336242*x + 1953011, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} - 45 x^{12} + 30 x^{11} + 1030 x^{10} + 632 x^{9} - 11085 x^{8} - 20276 x^{7} + 43102 x^{6} + 151674 x^{5} + 206418 x^{4} + 411952 x^{3} + 987997 x^{2} + 1336242 x + 1953011 \)

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:  \(-34413598978900358351795781632=-\,2^{21}\cdot 71^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $109.23$
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:  $2, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(568=2^{3}\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{568}(1,·)$, $\chi_{568}(387,·)$, $\chi_{568}(179,·)$, $\chi_{568}(385,·)$, $\chi_{568}(329,·)$, $\chi_{568}(427,·)$, $\chi_{568}(403,·)$, $\chi_{568}(321,·)$, $\chi_{568}(529,·)$, $\chi_{568}(91,·)$, $\chi_{568}(243,·)$, $\chi_{568}(545,·)$, $\chi_{568}(233,·)$, $\chi_{568}(187,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{6} + \frac{2}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{7} + \frac{2}{5} a^{5} + \frac{1}{5} a^{3} - \frac{1}{5} a$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2} - \frac{2}{5}$, $\frac{1}{85} a^{11} - \frac{2}{85} a^{10} - \frac{1}{85} a^{9} + \frac{3}{85} a^{8} - \frac{4}{17} a^{7} + \frac{23}{85} a^{6} + \frac{33}{85} a^{5} + \frac{1}{85} a^{4} + \frac{8}{17} a^{3} - \frac{2}{5} a^{2} - \frac{11}{85} a - \frac{2}{5}$, $\frac{1}{425} a^{12} + \frac{1}{425} a^{11} + \frac{2}{85} a^{10} + \frac{1}{25} a^{9} + \frac{6}{425} a^{8} - \frac{71}{425} a^{7} - \frac{8}{25} a^{6} + \frac{49}{425} a^{5} - \frac{8}{425} a^{4} + \frac{188}{425} a^{3} + \frac{91}{425} a^{2} + \frac{1}{425} a + \frac{11}{25}$, $\frac{1}{343070132086624660592873500225} a^{13} - \frac{15910591937235067167372384}{13722805283464986423714940009} a^{12} + \frac{708994264515218369457599369}{343070132086624660592873500225} a^{11} + \frac{6735257238241473519451016237}{343070132086624660592873500225} a^{10} + \frac{30517157165860945435049663959}{343070132086624660592873500225} a^{9} + \frac{32358966188892091425526937103}{343070132086624660592873500225} a^{8} - \frac{2611014512909398555714425365}{13722805283464986423714940009} a^{7} - \frac{17511812218694900624911175597}{68614026417324932118574700045} a^{6} + \frac{2850687930680208740362367858}{343070132086624660592873500225} a^{5} - \frac{115456028473802610192042087344}{343070132086624660592873500225} a^{4} + \frac{168267575827748158225066309408}{343070132086624660592873500225} a^{3} - \frac{11203876103538603552498234751}{68614026417324932118574700045} a^{2} + \frac{152709905909463068603049710421}{343070132086624660592873500225} a + \frac{8559961219398039076044230244}{20180596005095568270169029425}$

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

Class group and class number

$C_{1247}$, which has order $1247$ (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

\(\Q(\sqrt{-2}) \), 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 R ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$2$2.14.21.6$x^{14} + 4 x^{11} - 3 x^{10} + 4 x^{9} + 2 x^{8} + 2 x^{7} - 3 x^{6} + 2 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} - 2 x + 1$$2$$7$$21$$C_{14}$$[3]^{7}$
$71$71.14.12.1$x^{14} + 546629 x^{7} + 98234829011$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$