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Properties

Label 14.0.55977648821...4144.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 197^{12}$
Root discriminant $185.23$
Ramified primes $2, 197$
Class number $6328$ (GRH)
Class group $[2, 2, 1582]$ (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![1352569, 0, 11329577, 0, 20272254, 0, 3790966, 0, 284395, 0, 10186, 0, 169, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 169*x^12 + 10186*x^10 + 284395*x^8 + 3790966*x^6 + 20272254*x^4 + 11329577*x^2 + 1352569)
 
gp: K = bnfinit(x^14 + 169*x^12 + 10186*x^10 + 284395*x^8 + 3790966*x^6 + 20272254*x^4 + 11329577*x^2 + 1352569, 1)
 

Normalized defining polynomial

\( x^{14} + 169 x^{12} + 10186 x^{10} + 284395 x^{8} + 3790966 x^{6} + 20272254 x^{4} + 11329577 x^{2} + 1352569 \)

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:  \(-55977648821440226023583557894144=-\,2^{14}\cdot 197^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $185.23$
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:  $2, 197$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(788=2^{2}\cdot 197\)
Dirichlet character group:    $\lbrace$$\chi_{788}(1,·)$, $\chi_{788}(191,·)$, $\chi_{788}(395,·)$, $\chi_{788}(311,·)$, $\chi_{788}(705,·)$, $\chi_{788}(585,·)$, $\chi_{788}(375,·)$, $\chi_{788}(301,·)$, $\chi_{788}(627,·)$, $\chi_{788}(755,·)$, $\chi_{788}(695,·)$, $\chi_{788}(233,·)$, $\chi_{788}(769,·)$, $\chi_{788}(361,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{183328903859053111} a^{12} - \frac{14334329493014901}{183328903859053111} a^{10} + \frac{50036907682094334}{183328903859053111} a^{8} - \frac{86852188444878884}{183328903859053111} a^{6} + \frac{77831232864917001}{183328903859053111} a^{4} - \frac{81601665139696681}{183328903859053111} a^{2} + \frac{45190840461337261}{183328903859053111}$, $\frac{1}{213211515188078768093} a^{13} - \frac{65279424103315922417}{213211515188078768093} a^{11} + \frac{56148681488552346300}{213211515188078768093} a^{9} - \frac{53068905403711227963}{213211515188078768093} a^{7} - \frac{74903690445487805398}{213211515188078768093} a^{5} + \frac{98182690803312770815}{213211515188078768093} a^{3} + \frac{59077097883076439003}{213211515188078768093} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{1582}$, which has order $6328$ (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:  \( -\frac{40310893}{144316631450039} a^{13} - \frac{8389803843}{144316631450039} a^{11} - \frac{645004941341}{144316631450039} a^{9} - \frac{22736854425572}{144316631450039} a^{7} - \frac{369431084430812}{144316631450039} a^{5} - \frac{2278739116399986}{144316631450039} a^{3} - \frac{725438832932274}{144316631450039} a \) (order $4$)
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:  \( 1553055.199048291 \) (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

\(\Q(\sqrt{-1}) \), 7.7.58451728309129.1

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 ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$2$2.14.14.38$x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
$197$197.7.6.1$x^{7} - 197$$7$$1$$6$$C_7$$[\ ]_{7}$
197.7.6.1$x^{7} - 197$$7$$1$$6$$C_7$$[\ ]_{7}$