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Properties

Label 14.0.32908474225...8743.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{7}\cdot 43^{12}$
Root discriminant $66.48$
Ramified primes $7, 43$
Class number $448$ (GRH)
Class group $[2, 2, 2, 2, 2, 14]$ (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![85141, 87199, 53608, 3227, 5181, 8174, 6224, -171, -828, -82, 236, 45, -13, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 - 13*x^12 + 45*x^11 + 236*x^10 - 82*x^9 - 828*x^8 - 171*x^7 + 6224*x^6 + 8174*x^5 + 5181*x^4 + 3227*x^3 + 53608*x^2 + 87199*x + 85141)
 
gp: K = bnfinit(x^14 - 5*x^13 - 13*x^12 + 45*x^11 + 236*x^10 - 82*x^9 - 828*x^8 - 171*x^7 + 6224*x^6 + 8174*x^5 + 5181*x^4 + 3227*x^3 + 53608*x^2 + 87199*x + 85141, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} - 13 x^{12} + 45 x^{11} + 236 x^{10} - 82 x^{9} - 828 x^{8} - 171 x^{7} + 6224 x^{6} + 8174 x^{5} + 5181 x^{4} + 3227 x^{3} + 53608 x^{2} + 87199 x + 85141 \)

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:  \(-32908474225670013957008743=-\,7^{7}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.48$
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:  $7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(301=7\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{301}(64,·)$, $\chi_{301}(1,·)$, $\chi_{301}(293,·)$, $\chi_{301}(97,·)$, $\chi_{301}(41,·)$, $\chi_{301}(183,·)$, $\chi_{301}(78,·)$, $\chi_{301}(176,·)$, $\chi_{301}(274,·)$, $\chi_{301}(279,·)$, $\chi_{301}(216,·)$, $\chi_{301}(90,·)$, $\chi_{301}(188,·)$, $\chi_{301}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{9} + \frac{2}{7} a^{8} + \frac{2}{7} a^{7} - \frac{2}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2}$, $\frac{1}{7} a^{11} + \frac{1}{7} a^{9} - \frac{3}{7} a^{8} + \frac{2}{7} a^{6} - \frac{2}{7} a^{5} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2}$, $\frac{1}{49} a^{12} - \frac{3}{49} a^{11} + \frac{3}{49} a^{10} - \frac{15}{49} a^{9} + \frac{13}{49} a^{8} + \frac{20}{49} a^{7} + \frac{9}{49} a^{6} + \frac{2}{7} a^{5} + \frac{10}{49} a^{4} + \frac{18}{49} a^{3} - \frac{16}{49} a^{2} - \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{9162732156031460760179027} a^{13} + \frac{60049304364567196641990}{9162732156031460760179027} a^{12} - \frac{568245322517319971570357}{9162732156031460760179027} a^{11} + \frac{566736201404098500651567}{9162732156031460760179027} a^{10} - \frac{725946752661721953242156}{9162732156031460760179027} a^{9} + \frac{870346047319418169305075}{9162732156031460760179027} a^{8} - \frac{520472158453063649785144}{1308961736575922965739861} a^{7} - \frac{731362843500840771164423}{9162732156031460760179027} a^{6} + \frac{1420377716508228303129895}{9162732156031460760179027} a^{5} + \frac{4038258691115300880931811}{9162732156031460760179027} a^{4} + \frac{2197075755506815069517462}{9162732156031460760179027} a^{3} + \frac{2762101747127436307038595}{9162732156031460760179027} a^{2} - \frac{384942408242401782154658}{1308961736575922965739861} a - \frac{406487325636935299657006}{1308961736575922965739861}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{14}$, which has order $448$ (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:  \( 35991.64185055774 \) (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

\(\Q(\sqrt{-7}) \), 7.7.6321363049.1

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 ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ R ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
$43$43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$