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Properties

Label 14.14.9830707784...2272.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{16}\cdot 7^{14}\cdot 13^{12}\cdot 37^{7}$
Root discriminant $847.31$
Ramified primes $2, 7, 13, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T37

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![53095268772, 33962524596, -11527474959, -10031125468, 178767862, 960908312, 74520719, -37791624, -4735640, 567476, 94367, -1820, -546, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 546*x^12 - 1820*x^11 + 94367*x^10 + 567476*x^9 - 4735640*x^8 - 37791624*x^7 + 74520719*x^6 + 960908312*x^5 + 178767862*x^4 - 10031125468*x^3 - 11527474959*x^2 + 33962524596*x + 53095268772)
 
gp: K = bnfinit(x^14 - 546*x^12 - 1820*x^11 + 94367*x^10 + 567476*x^9 - 4735640*x^8 - 37791624*x^7 + 74520719*x^6 + 960908312*x^5 + 178767862*x^4 - 10031125468*x^3 - 11527474959*x^2 + 33962524596*x + 53095268772, 1)
 

Normalized defining polynomial

\( x^{14} - 546 x^{12} - 1820 x^{11} + 94367 x^{10} + 567476 x^{9} - 4735640 x^{8} - 37791624 x^{7} + 74520719 x^{6} + 960908312 x^{5} + 178767862 x^{4} - 10031125468 x^{3} - 11527474959 x^{2} + 33962524596 x + 53095268772 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(98307077841097735657969434524441383862272=2^{16}\cdot 7^{14}\cdot 13^{12}\cdot 37^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $847.31$
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:  $2, 7, 13, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{5} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} + \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{208} a^{7} - \frac{1}{16} a^{5} - \frac{1}{8} a^{4} - \frac{1}{16} a^{3} - \frac{3}{16} a + \frac{1}{8}$, $\frac{1}{208} a^{8} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{2} + \frac{1}{8} a$, $\frac{1}{416} a^{9} - \frac{1}{16} a^{6} + \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{1}{4} a^{3} - \frac{3}{16} a^{2} + \frac{1}{32} a + \frac{5}{16}$, $\frac{1}{832} a^{10} - \frac{1}{832} a^{9} - \frac{1}{416} a^{8} - \frac{1}{32} a^{6} + \frac{3}{32} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{27}{64} a^{2} - \frac{9}{64} a - \frac{11}{32}$, $\frac{1}{5824} a^{11} + \frac{1}{2912} a^{10} + \frac{5}{5824} a^{9} - \frac{5}{2912} a^{8} - \frac{5}{2912} a^{7} - \frac{3}{32} a^{5} - \frac{5}{64} a^{3} - \frac{11}{32} a^{2} + \frac{23}{64} a - \frac{9}{32}$, $\frac{1}{17472} a^{12} - \frac{1}{2912} a^{10} + \frac{1}{17472} a^{9} - \frac{1}{4368} a^{8} + \frac{5}{4368} a^{7} + \frac{1}{48} a^{6} - \frac{1}{32} a^{5} - \frac{13}{192} a^{4} - \frac{1}{48} a^{3} + \frac{11}{96} a^{2} + \frac{59}{192} a + \frac{5}{32}$, $\frac{1}{365992544096993991951655961159808} a^{13} + \frac{2329743098762551348812524215}{121997514698997997317218653719936} a^{12} - \frac{9838592928720016667035308515}{121997514698997997317218653719936} a^{11} + \frac{132025912442963300168291454127}{365992544096993991951655961159808} a^{10} + \frac{198151127748384829132652700163}{182996272048496995975827980579904} a^{9} - \frac{355382902936413530875149246137}{182996272048496995975827980579904} a^{8} - \frac{30378506707187501499061249339}{26142324578356713710832568654272} a^{7} + \frac{2611033620820456733337852441}{223438671609886441972927937216} a^{6} + \frac{478901395080364456934820823967}{4021896088977955955512702869888} a^{5} - \frac{29320762993880888645552606437}{4021896088977955955512702869888} a^{4} - \frac{570940867925412977922936044279}{4021896088977955955512702869888} a^{3} - \frac{636964781854947678589306829167}{4021896088977955955512702869888} a^{2} + \frac{251437384555154379121496693}{1209956705468699144257732512} a + \frac{21061733691011071810003750843}{111719335804943220986463968608}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 29462583765900000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T37:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1764
The 25 conjugacy class representatives for [1/2.F_42(7)^2]2
Character table for [1/2.F_42(7)^2]2 is not computed

Intermediate fields

\(\Q(\sqrt{37}) \)

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

Sibling fields

Degree 21 siblings: data not computed
Degree 28 sibling: data not computed
Degree 42 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.7.0.1}{7} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.12.16.3$x^{12} - 30 x^{10} - 5 x^{8} + 19 x^{4} + 30 x^{2} + 1$$6$$2$$16$$C_6\times S_3$$[2]_{3}^{6}$
$7$7.7.7.3$x^{7} + 35 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
7.7.7.2$x^{7} + 21 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
13Data not computed
37Data not computed