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Properties

Label 14.0.19695924271...5728.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 7^{24}\cdot 13^{7}$
Root discriminant $202.65$
Ramified primes $2, 7, 13$
Class number $1217566$ (GRH)
Class group $[1217566]$ (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![665398601, 135111620, 161055601, 21707980, 18366768, 1649242, 1275554, 56436, 59423, 560, 1988, -42, 49, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 49*x^12 - 42*x^11 + 1988*x^10 + 560*x^9 + 59423*x^8 + 56436*x^7 + 1275554*x^6 + 1649242*x^5 + 18366768*x^4 + 21707980*x^3 + 161055601*x^2 + 135111620*x + 665398601)
 
gp: K = bnfinit(x^14 + 49*x^12 - 42*x^11 + 1988*x^10 + 560*x^9 + 59423*x^8 + 56436*x^7 + 1275554*x^6 + 1649242*x^5 + 18366768*x^4 + 21707980*x^3 + 161055601*x^2 + 135111620*x + 665398601, 1)
 

Normalized defining polynomial

\( x^{14} + 49 x^{12} - 42 x^{11} + 1988 x^{10} + 560 x^{9} + 59423 x^{8} + 56436 x^{7} + 1275554 x^{6} + 1649242 x^{5} + 18366768 x^{4} + 21707980 x^{3} + 161055601 x^{2} + 135111620 x + 665398601 \)

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:  \(-196959242717829613546212447305728=-\,2^{14}\cdot 7^{24}\cdot 13^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $202.65$
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:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2548=2^{2}\cdot 7^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2548}(1,·)$, $\chi_{2548}(2339,·)$, $\chi_{2548}(1093,·)$, $\chi_{2548}(519,·)$, $\chi_{2548}(2185,·)$, $\chi_{2548}(1611,·)$, $\chi_{2548}(365,·)$, $\chi_{2548}(1457,·)$, $\chi_{2548}(883,·)$, $\chi_{2548}(1975,·)$, $\chi_{2548}(729,·)$, $\chi_{2548}(155,·)$, $\chi_{2548}(1821,·)$, $\chi_{2548}(1247,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{589} a^{11} + \frac{256}{589} a^{10} - \frac{288}{589} a^{9} - \frac{196}{589} a^{8} + \frac{145}{589} a^{7} - \frac{53}{589} a^{6} + \frac{67}{589} a^{5} - \frac{5}{19} a^{4} + \frac{24}{589} a^{3} + \frac{147}{589} a^{2} + \frac{89}{589} a$, $\frac{1}{589} a^{12} + \frac{144}{589} a^{10} - \frac{3}{19} a^{9} + \frac{256}{589} a^{8} - \frac{66}{589} a^{7} + \frac{88}{589} a^{6} - \frac{226}{589} a^{5} + \frac{241}{589} a^{4} - \frac{107}{589} a^{3} + \frac{153}{589} a^{2} + \frac{187}{589} a$, $\frac{1}{100064144588917322288429718382340131829} a^{13} - \frac{67391510188981266370188146709169024}{100064144588917322288429718382340131829} a^{12} - \frac{25379735691257920718952114291837002}{100064144588917322288429718382340131829} a^{11} - \frac{16943043369072060417142855831406981396}{100064144588917322288429718382340131829} a^{10} - \frac{41592549531586628451237593095465275875}{100064144588917322288429718382340131829} a^{9} + \frac{96827109564177575333770244134317334}{5266533925732490646759458862228427991} a^{8} + \frac{44311792169069126838713269112693922015}{100064144588917322288429718382340131829} a^{7} - \frac{5652521391001236396102825121146304776}{100064144588917322288429718382340131829} a^{6} + \frac{40755702615673772771057049101434342479}{100064144588917322288429718382340131829} a^{5} + \frac{45187162005929280318437461124951260755}{100064144588917322288429718382340131829} a^{4} - \frac{17971517916960221230489412729642589851}{100064144588917322288429718382340131829} a^{3} + \frac{32906486519937972143690731488609565196}{100064144588917322288429718382340131829} a^{2} + \frac{39391170514755867932193758613130244324}{100064144588917322288429718382340131829} a + \frac{54821368662094453263388042548804955}{169888191152660988605143834265433161}$

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

Class group and class number

$C_{1217566}$, which has order $1217566$ (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:  \( 35256.68973693789 \) (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

\(\Q(\sqrt{-13}) \), 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 ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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$2.14.14.15$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}$$[2]^{7}$
$7$7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
$13$13.14.7.1$x^{14} - 43940 x^{8} + 482680900 x^{2} - 250994068$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$