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Properties

Label 14.0.57969014080...7744.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 29^{12}$
Root discriminant $35.85$
Ramified primes $2, 29$
Class number $56$
Class group $[2, 2, 14]$
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, 0, 53, 0, 686, 0, 1194, 0, 767, 0, 214, 0, 25, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 25*x^12 + 214*x^10 + 767*x^8 + 1194*x^6 + 686*x^4 + 53*x^2 + 1)
 
gp: K = bnfinit(x^14 + 25*x^12 + 214*x^10 + 767*x^8 + 1194*x^6 + 686*x^4 + 53*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{14} + 25 x^{12} + 214 x^{10} + 767 x^{8} + 1194 x^{6} + 686 x^{4} + 53 x^{2} + 1 \)

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:  \(-5796901408038404767744=-\,2^{14}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.85$
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:  $2, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(116=2^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{116}(1,·)$, $\chi_{116}(107,·)$, $\chi_{116}(81,·)$, $\chi_{116}(65,·)$, $\chi_{116}(103,·)$, $\chi_{116}(45,·)$, $\chi_{116}(111,·)$, $\chi_{116}(49,·)$, $\chi_{116}(83,·)$, $\chi_{116}(53,·)$, $\chi_{116}(23,·)$, $\chi_{116}(7,·)$, $\chi_{116}(25,·)$, $\chi_{116}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{17} a^{10} + \frac{7}{17} a^{8} + \frac{7}{17} a^{6} + \frac{6}{17} a^{4} - \frac{8}{17} a^{2} + \frac{4}{17}$, $\frac{1}{17} a^{11} + \frac{7}{17} a^{9} + \frac{7}{17} a^{7} + \frac{6}{17} a^{5} - \frac{8}{17} a^{3} + \frac{4}{17} a$, $\frac{1}{11849} a^{12} - \frac{52}{11849} a^{10} + \frac{4218}{11849} a^{8} - \frac{4096}{11849} a^{6} - \frac{3337}{11849} a^{4} - \frac{179}{697} a^{2} - \frac{2616}{11849}$, $\frac{1}{11849} a^{13} - \frac{52}{11849} a^{11} + \frac{4218}{11849} a^{9} - \frac{4096}{11849} a^{7} - \frac{3337}{11849} a^{5} - \frac{179}{697} a^{3} - \frac{2616}{11849} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{14}$, which has order $56$

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:  \( \frac{1059}{11849} a^{13} + \frac{26481}{11849} a^{11} + \frac{227011}{11849} a^{9} + \frac{818742}{11849} a^{7} + \frac{1303994}{11849} a^{5} + \frac{47378}{697} a^{3} + \frac{91538}{11849} a \) (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
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

\(\Q(\sqrt{-1}) \), 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 R ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$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}$$[2]^{7}$
$29$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$