↑

Properties

Label 14.0.59549071109...5271.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{7}\cdot 337^{13}$
Root discriminant $588.35$
Ramified primes $7, 337$
Class number $2985668$ (GRH)
Class group $[2985668]$ (GRH)
Galois group $C_{14}$ (as 14T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![757415595679, 365724334139, 232439703661, 37382582628, 11573214816, -1273665671, 317860920, -50472370, 8805688, -520762, 102790, -1833, 518, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 518*x^12 - 1833*x^11 + 102790*x^10 - 520762*x^9 + 8805688*x^8 - 50472370*x^7 + 317860920*x^6 - 1273665671*x^5 + 11573214816*x^4 + 37382582628*x^3 + 232439703661*x^2 + 365724334139*x + 757415595679)
 
gp: K = bnfinit(x^14 - x^13 + 518*x^12 - 1833*x^11 + 102790*x^10 - 520762*x^9 + 8805688*x^8 - 50472370*x^7 + 317860920*x^6 - 1273665671*x^5 + 11573214816*x^4 + 37382582628*x^3 + 232439703661*x^2 + 365724334139*x + 757415595679, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 518 x^{12} - 1833 x^{11} + 102790 x^{10} - 520762 x^{9} + 8805688 x^{8} - 50472370 x^{7} + 317860920 x^{6} - 1273665671 x^{5} + 11573214816 x^{4} + 37382582628 x^{3} + 232439703661 x^{2} + 365724334139 x + 757415595679 \)

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:  \(-595490711098343828041968400200063015271=-\,7^{7}\cdot 337^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $588.35$
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:  $7, 337$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2359=7\cdot 337\)
Dirichlet character group:    $\lbrace$$\chi_{2359}(512,·)$, $\chi_{2359}(1,·)$, $\chi_{2359}(258,·)$, $\chi_{2359}(295,·)$, $\chi_{2359}(8,·)$, $\chi_{2359}(64,·)$, $\chi_{2359}(2295,·)$, $\chi_{2359}(622,·)$, $\chi_{2359}(2351,·)$, $\chi_{2359}(2064,·)$, $\chi_{2359}(1847,·)$, $\chi_{2359}(2101,·)$, $\chi_{2359}(2358,·)$, $\chi_{2359}(1737,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{55657576968377806953723067618236774051900928540347952761119886814874294837} a^{13} - \frac{5956523599322580243264949025124982500201691731809494563546290807963730866}{55657576968377806953723067618236774051900928540347952761119886814874294837} a^{12} + \frac{17331365637002940419544986206603877363701770536915558419573631045423558460}{55657576968377806953723067618236774051900928540347952761119886814874294837} a^{11} + \frac{16786726961506608984326904472240220973958670738969041662604019593410453925}{55657576968377806953723067618236774051900928540347952761119886814874294837} a^{10} - \frac{4715042873447171705540884848678468613789075567814538871872842657317993819}{55657576968377806953723067618236774051900928540347952761119886814874294837} a^{9} + \frac{2849864462890581759545049679815502860627356270453551717590185831411738651}{55657576968377806953723067618236774051900928540347952761119886814874294837} a^{8} - \frac{14210264696219808987037669856536954208134807532518295396427962738540700740}{55657576968377806953723067618236774051900928540347952761119886814874294837} a^{7} - \frac{23468391461939134568416552947473928962892999636689775902459008481253228150}{55657576968377806953723067618236774051900928540347952761119886814874294837} a^{6} - \frac{15858816147346342335889942543142331856465615863821519001915155591991241204}{55657576968377806953723067618236774051900928540347952761119886814874294837} a^{5} - \frac{14486997350897276146645117939838768570012103602561154339260095272194174653}{55657576968377806953723067618236774051900928540347952761119886814874294837} a^{4} + \frac{13608467845858425413742216279286094795775531778249587164309197177108965792}{55657576968377806953723067618236774051900928540347952761119886814874294837} a^{3} - \frac{412466292833077941844936040155370705933828082841135137170228792656169965}{55657576968377806953723067618236774051900928540347952761119886814874294837} a^{2} + \frac{1860541459993440595164791540107651407352890599231392842269205901770429906}{55657576968377806953723067618236774051900928540347952761119886814874294837} a - \frac{22966925924533832861230448450290190729169063575859602911877225698287440774}{55657576968377806953723067618236774051900928540347952761119886814874294837}$

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

Class group and class number

$C_{2985668}$, which has order $2985668$ (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:  \( 5486180.267403393 \) (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

\(\Q(\sqrt{-2359}) \), 7.7.1464803622199009.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{14}$

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
$7$7.14.7.1$x^{14} - 117649 x^{2} + 1647086$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
337Data not computed