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Properties

Label 14.0.77379293087...2667.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 29^{12}$
Root discriminant $31.05$
Ramified primes $3, 29$
Class number $8$
Class group $[2, 2, 2]$
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![1, 9, 67, 182, 455, -254, 776, -132, 353, -42, 123, -2, 13, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 13*x^12 - 2*x^11 + 123*x^10 - 42*x^9 + 353*x^8 - 132*x^7 + 776*x^6 - 254*x^5 + 455*x^4 + 182*x^3 + 67*x^2 + 9*x + 1)
 
gp: K = bnfinit(x^14 - x^13 + 13*x^12 - 2*x^11 + 123*x^10 - 42*x^9 + 353*x^8 - 132*x^7 + 776*x^6 - 254*x^5 + 455*x^4 + 182*x^3 + 67*x^2 + 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 13 x^{12} - 2 x^{11} + 123 x^{10} - 42 x^{9} + 353 x^{8} - 132 x^{7} + 776 x^{6} - 254 x^{5} + 455 x^{4} + 182 x^{3} + 67 x^{2} + 9 x + 1 \)

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:  \(-773792930870360792667=-\,3^{7}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.05$
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:  $3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(87=3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{87}(1,·)$, $\chi_{87}(65,·)$, $\chi_{87}(74,·)$, $\chi_{87}(7,·)$, $\chi_{87}(16,·)$, $\chi_{87}(49,·)$, $\chi_{87}(82,·)$, $\chi_{87}(83,·)$, $\chi_{87}(20,·)$, $\chi_{87}(53,·)$, $\chi_{87}(23,·)$, $\chi_{87}(25,·)$, $\chi_{87}(59,·)$, $\chi_{87}(52,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17} a^{12} + \frac{2}{17} a^{11} - \frac{7}{17} a^{10} - \frac{7}{17} a^{9} - \frac{5}{17} a^{8} + \frac{6}{17} a^{7} + \frac{8}{17} a^{6} + \frac{8}{17} a^{5} - \frac{3}{17} a^{4} + \frac{5}{17} a^{3} + \frac{4}{17} a^{2} - \frac{4}{17} a + \frac{2}{17}$, $\frac{1}{171721640178611} a^{13} + \frac{3434584787440}{171721640178611} a^{12} - \frac{83474705599767}{171721640178611} a^{11} + \frac{36253161028700}{171721640178611} a^{10} - \frac{40655906740414}{171721640178611} a^{9} - \frac{83347585304740}{171721640178611} a^{8} + \frac{1890101463184}{171721640178611} a^{7} - \frac{21506824520547}{171721640178611} a^{6} + \frac{54693714322909}{171721640178611} a^{5} + \frac{25985495782295}{171721640178611} a^{4} + \frac{51706636683181}{171721640178611} a^{3} - \frac{3210106905605}{10101272951683} a^{2} + \frac{42018727146718}{171721640178611} a - \frac{59720899670656}{171721640178611}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$

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:  \( -\frac{1211436584386}{10101272951683} a^{13} + \frac{1268726932500}{10101272951683} a^{12} - \frac{15853046937908}{10101272951683} a^{11} + \frac{3237618802589}{10101272951683} a^{10} - \frac{149741593511519}{10101272951683} a^{9} + \frac{58298011189649}{10101272951683} a^{8} - \frac{435698640216490}{10101272951683} a^{7} + \frac{184864417677595}{10101272951683} a^{6} - \frac{963600075755546}{10101272951683} a^{5} + \frac{365423238302894}{10101272951683} a^{4} - \frac{601398619128417}{10101272951683} a^{3} - \frac{164222117360647}{10101272951683} a^{2} - \frac{89616122106983}{10101272951683} a - \frac{1950913215886}{10101272951683} \) (order $6$)
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:  \( 6020.98510015 \)
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

\(\Q(\sqrt{-3}) \), 7.7.594823321.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$