# Properties

 Label 14.0.16811014083...4576.1 Degree $14$ Signature $[0, 7]$ Discriminant $-\,2^{14}\cdot 29^{13}$ Root discriminant $45.60$ Ramified primes $2, 29$ Class number $48$ (GRH) Class group $[2, 2, 2, 6]$ (GRH) Galois group $C_{14}$ (as 14T1)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 29*x^12 + 290*x^10 + 1247*x^8 + 2262*x^6 + 1566*x^4 + 377*x^2 + 29)

gp: K = bnfinit(x^14 + 29*x^12 + 290*x^10 + 1247*x^8 + 2262*x^6 + 1566*x^4 + 377*x^2 + 29, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29, 0, 377, 0, 1566, 0, 2262, 0, 1247, 0, 290, 0, 29, 0, 1]);

## Normalizeddefining polynomial

$$x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $14$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 7]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-168110140833113738264576=-\,2^{14}\cdot 29^{13}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $45.60$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $2, 29$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $14$ This field is Galois and abelian over $\Q$. Conductor: $$116=2^{2}\cdot 29$$ Dirichlet character group: $\lbrace$$\chi_{116}(1,·), \chi_{116}(81,·), \chi_{116}(67,·), \chi_{116}(51,·), \chi_{116}(65,·), \chi_{116}(71,·), \chi_{116}(45,·), \chi_{116}(49,·), \chi_{116}(115,·), \chi_{116}(53,·), \chi_{116}(25,·), \chi_{116}(91,·), \chi_{116}(35,·), \chi_{116}(63,·)$$\rbrace$ This is a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{17} a^{10} - \frac{5}{17} a^{8} + \frac{3}{17} a^{6} - \frac{4}{17} a^{4} + \frac{7}{17} a^{2} - \frac{6}{17}$, $\frac{1}{17} a^{11} - \frac{5}{17} a^{9} + \frac{3}{17} a^{7} - \frac{4}{17} a^{5} + \frac{7}{17} a^{3} - \frac{6}{17} a$, $\frac{1}{41123} a^{12} - \frac{75}{41123} a^{10} + \frac{15347}{41123} a^{8} + \frac{13794}{41123} a^{6} - \frac{12361}{41123} a^{4} - \frac{16731}{41123} a^{2} - \frac{18212}{41123}$, $\frac{1}{41123} a^{13} - \frac{75}{41123} a^{11} + \frac{15347}{41123} a^{9} + \frac{13794}{41123} a^{7} - \frac{12361}{41123} a^{5} - \frac{16731}{41123} a^{3} - \frac{18212}{41123} a$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}$, which has order $48$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $6$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); 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) sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$6020.98510015$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$C_{14}$ (as 14T1):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A cyclic group of order 14 The 14 conjugacy class representatives for $C_{14}$ Character table for $C_{14}$

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])

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];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7} 2929.14.13.1x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.116.2t1.a.a$1$ $2^{2} \cdot 29$ $x^{2} + 29$ $C_2$ (as 2T1) $1$ $-1$
* 1.29.7t1.a.a$1$ $29$ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.116.14t1.a.a$1$ $2^{2} \cdot 29$ $x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.a.b$1$ $29$ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.116.14t1.a.b$1$ $2^{2} \cdot 29$ $x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.a.c$1$ $29$ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.116.14t1.a.c$1$ $2^{2} \cdot 29$ $x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.a.d$1$ $29$ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.116.14t1.a.d$1$ $2^{2} \cdot 29$ $x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.a.e$1$ $29$ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.116.14t1.a.e$1$ $2^{2} \cdot 29$ $x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.a.f$1$ $29$ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.116.14t1.a.f$1$ $2^{2} \cdot 29$ $x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$ $C_{14}$ (as 14T1) $0$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.