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Properties

Label 14.0.31388668949...5984.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 7^{24}$
Root discriminant $56.20$
Ramified primes $2, 7$
Class number $71$ (GRH)
Class group $[71]$ (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![9409, 0, 28784, 0, 31906, 0, 16513, 0, 4431, 0, 623, 0, 42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 42*x^12 + 623*x^10 + 4431*x^8 + 16513*x^6 + 31906*x^4 + 28784*x^2 + 9409)
 
gp: K = bnfinit(x^14 + 42*x^12 + 623*x^10 + 4431*x^8 + 16513*x^6 + 31906*x^4 + 28784*x^2 + 9409, 1)
 

Normalized defining polynomial

\( x^{14} + 42 x^{12} + 623 x^{10} + 4431 x^{8} + 16513 x^{6} + 31906 x^{4} + 28784 x^{2} + 9409 \)

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:  \(-3138866894939200133545984=-\,2^{14}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.20$
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:  $2, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(196=2^{2}\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{196}(1,·)$, $\chi_{196}(99,·)$, $\chi_{196}(71,·)$, $\chi_{196}(169,·)$, $\chi_{196}(43,·)$, $\chi_{196}(141,·)$, $\chi_{196}(15,·)$, $\chi_{196}(113,·)$, $\chi_{196}(85,·)$, $\chi_{196}(183,·)$, $\chi_{196}(57,·)$, $\chi_{196}(155,·)$, $\chi_{196}(29,·)$, $\chi_{196}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19} a^{10} + \frac{9}{19} a^{8} - \frac{8}{19} a^{6} - \frac{2}{19} a^{4} + \frac{4}{19} a^{2} + \frac{9}{19}$, $\frac{1}{19} a^{11} + \frac{9}{19} a^{9} - \frac{8}{19} a^{7} - \frac{2}{19} a^{5} + \frac{4}{19} a^{3} + \frac{9}{19} a$, $\frac{1}{884089} a^{12} + \frac{3588}{884089} a^{10} + \frac{11175}{28519} a^{8} + \frac{427860}{884089} a^{6} + \frac{111349}{884089} a^{4} - \frac{312323}{884089} a^{2} + \frac{294943}{884089}$, $\frac{1}{85756633} a^{13} + \frac{1957890}{85756633} a^{11} + \frac{492996}{2766343} a^{9} + \frac{29881983}{85756633} a^{7} + \frac{5043635}{85756633} a^{5} - \frac{34047298}{85756633} a^{3} - \frac{31625323}{85756633} a$

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

Class group and class number

$C_{71}$, which has order $71$ (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:  \( \frac{31801}{2766343} a^{13} + \frac{1214780}{2766343} a^{11} + \frac{15233526}{2766343} a^{9} + \frac{84420729}{2766343} a^{7} + \frac{220503353}{2766343} a^{5} + \frac{253711850}{2766343} a^{3} + \frac{101685766}{2766343} a \) (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:  \( 35256.6897369 \) (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

\(\Q(\sqrt{-1}) \), 7.7.13841287201.1

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 ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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}$
7Data not computed