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Properties

Label 14.14.3757843639...6441.1
Degree $14$
Signature $[14, 0]$
Discriminant $3^{7}\cdot 43^{13}$
Root discriminant $56.93$
Ramified primes $3, 43$
Class number $1$
Class group Trivial
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![337, -434, -4323, -45, 16761, 19387, 1803, -7649, -2996, 887, 585, -19, -41, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 - 41*x^12 - 19*x^11 + 585*x^10 + 887*x^9 - 2996*x^8 - 7649*x^7 + 1803*x^6 + 19387*x^5 + 16761*x^4 - 45*x^3 - 4323*x^2 - 434*x + 337)
 
gp: K = bnfinit(x^14 - x^13 - 41*x^12 - 19*x^11 + 585*x^10 + 887*x^9 - 2996*x^8 - 7649*x^7 + 1803*x^6 + 19387*x^5 + 16761*x^4 - 45*x^3 - 4323*x^2 - 434*x + 337, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} - 41 x^{12} - 19 x^{11} + 585 x^{10} + 887 x^{9} - 2996 x^{8} - 7649 x^{7} + 1803 x^{6} + 19387 x^{5} + 16761 x^{4} - 45 x^{3} - 4323 x^{2} - 434 x + 337 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3757843639805369947326441=3^{7}\cdot 43^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.93$
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:  $3, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(129=3\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{129}(128,·)$, $\chi_{129}(1,·)$, $\chi_{129}(2,·)$, $\chi_{129}(4,·)$, $\chi_{129}(65,·)$, $\chi_{129}(8,·)$, $\chi_{129}(64,·)$, $\chi_{129}(32,·)$, $\chi_{129}(16,·)$, $\chi_{129}(113,·)$, $\chi_{129}(97,·)$, $\chi_{129}(121,·)$, $\chi_{129}(125,·)$, $\chi_{129}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{22021263193} a^{13} - \frac{5111742488}{22021263193} a^{12} - \frac{10553547164}{22021263193} a^{11} + \frac{6473991036}{22021263193} a^{10} - \frac{9862863353}{22021263193} a^{9} + \frac{3543158665}{22021263193} a^{8} + \frac{544668076}{22021263193} a^{7} - \frac{5156780828}{22021263193} a^{6} + \frac{6540340740}{22021263193} a^{5} + \frac{10330000822}{22021263193} a^{4} - \frac{8369430043}{22021263193} a^{3} + \frac{3854848764}{22021263193} a^{2} + \frac{6442325060}{22021263193} a - \frac{3029595438}{22021263193}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 38725608.4697 \)
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

\(\Q(\sqrt{129}) \), 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ R ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$3$3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$43$43.14.13.11$x^{14} + 205667667$$14$$1$$13$$C_{14}$$[\ ]_{14}$