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Properties

Label 14.0.30278124828...3616.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 3^{7}\cdot 7^{7}\cdot 29^{13}$
Root discriminant $208.97$
Ramified primes $2, 3, 7, 29$
Class number $2108416$ (GRH)
Class group $[2, 2, 4, 4, 4, 8236]$ (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![52231567689, 0, 32333827617, 0, 6395702166, 0, 439916022, 0, 11548467, 0, 127890, 0, 609, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 609*x^12 + 127890*x^10 + 11548467*x^8 + 439916022*x^6 + 6395702166*x^4 + 32333827617*x^2 + 52231567689)
 
gp: K = bnfinit(x^14 + 609*x^12 + 127890*x^10 + 11548467*x^8 + 439916022*x^6 + 6395702166*x^4 + 32333827617*x^2 + 52231567689, 1)
 

Normalized defining polynomial

\( x^{14} + 609 x^{12} + 127890 x^{10} + 11548467 x^{8} + 439916022 x^{6} + 6395702166 x^{4} + 32333827617 x^{2} + 52231567689 \)

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:  \(-302781248280417347338001059823616=-\,2^{14}\cdot 3^{7}\cdot 7^{7}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $208.97$
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:  $2, 3, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2436=2^{2}\cdot 3\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{2436}(1,·)$, $\chi_{2436}(2435,·)$, $\chi_{2436}(1093,·)$, $\chi_{2436}(2017,·)$, $\chi_{2436}(169,·)$, $\chi_{2436}(167,·)$, $\chi_{2436}(1427,·)$, $\chi_{2436}(1009,·)$, $\chi_{2436}(419,·)$, $\chi_{2436}(2267,·)$, $\chi_{2436}(1343,·)$, $\chi_{2436}(671,·)$, $\chi_{2436}(2269,·)$, $\chi_{2436}(1765,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{21} a^{2}$, $\frac{1}{21} a^{3}$, $\frac{1}{441} a^{4}$, $\frac{1}{441} a^{5}$, $\frac{1}{9261} a^{6}$, $\frac{1}{9261} a^{7}$, $\frac{1}{194481} a^{8}$, $\frac{1}{194481} a^{9}$, $\frac{1}{69429717} a^{10} - \frac{5}{3306177} a^{8} + \frac{1}{52479} a^{6} - \frac{4}{7497} a^{4} + \frac{1}{51} a^{2} - \frac{6}{17}$, $\frac{1}{69429717} a^{11} - \frac{5}{3306177} a^{9} + \frac{1}{52479} a^{7} - \frac{4}{7497} a^{5} + \frac{1}{51} a^{3} - \frac{6}{17} a$, $\frac{1}{3526960193883} a^{12} - \frac{25}{55983495141} a^{10} + \frac{15347}{7997642163} a^{8} + \frac{4598}{126946701} a^{6} - \frac{12361}{18135243} a^{4} - \frac{5577}{287861} a^{2} - \frac{18212}{41123}$, $\frac{1}{3526960193883} a^{13} - \frac{25}{55983495141} a^{11} + \frac{15347}{7997642163} a^{9} + \frac{4598}{126946701} a^{7} - \frac{12361}{18135243} a^{5} - \frac{5577}{287861} a^{3} - \frac{18212}{41123} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{8236}$, which has order $2108416$ (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:  \( -1 \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6020.985100147561 \) (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

\(\Q(\sqrt{-609}) \), 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 R ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$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}$$[2]^{7}$
$3$3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$7$7.14.7.2$x^{14} - 686 x^{8} + 117649 x^{2} - 3294172$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.13.11$x^{14} + 3712$$14$$1$$13$$C_{14}$$[\ ]_{14}$