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Properties

Label 14.0.20984256997...9051.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 7^{7}\cdot 71^{13}$
Root discriminant $239.96$
Ramified primes $3, 7, 71$
Class number $2093964$ (GRH)
Class group $[2, 1046982]$ (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![4370228125, 1543186750, 2982769075, 560196080, 499919527, 54051465, 39941900, 1571901, 1703750, 6901, 40159, -366, 358, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 358*x^12 - 366*x^11 + 40159*x^10 + 6901*x^9 + 1703750*x^8 + 1571901*x^7 + 39941900*x^6 + 54051465*x^5 + 499919527*x^4 + 560196080*x^3 + 2982769075*x^2 + 1543186750*x + 4370228125)
 
gp: K = bnfinit(x^14 - x^13 + 358*x^12 - 366*x^11 + 40159*x^10 + 6901*x^9 + 1703750*x^8 + 1571901*x^7 + 39941900*x^6 + 54051465*x^5 + 499919527*x^4 + 560196080*x^3 + 2982769075*x^2 + 1543186750*x + 4370228125, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 358 x^{12} - 366 x^{11} + 40159 x^{10} + 6901 x^{9} + 1703750 x^{8} + 1571901 x^{7} + 39941900 x^{6} + 54051465 x^{5} + 499919527 x^{4} + 560196080 x^{3} + 2982769075 x^{2} + 1543186750 x + 4370228125 \)

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:  \(-2098425699738568435088694126099051=-\,3^{7}\cdot 7^{7}\cdot 71^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $239.96$
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:  $3, 7, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1491=3\cdot 7\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{1491}(1,·)$, $\chi_{1491}(1091,·)$, $\chi_{1491}(1028,·)$, $\chi_{1491}(41,·)$, $\chi_{1491}(1450,·)$, $\chi_{1491}(463,·)$, $\chi_{1491}(400,·)$, $\chi_{1491}(1490,·)$, $\chi_{1491}(1301,·)$, $\chi_{1491}(1175,·)$, $\chi_{1491}(1156,·)$, $\chi_{1491}(335,·)$, $\chi_{1491}(316,·)$, $\chi_{1491}(190,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{25} a^{8} + \frac{1}{25} a^{7} - \frac{2}{25} a^{6} - \frac{6}{25} a^{4} + \frac{4}{25} a^{3} - \frac{3}{25} a^{2} + \frac{1}{5} a$, $\frac{1}{25} a^{9} + \frac{2}{25} a^{7} + \frac{2}{25} a^{6} - \frac{1}{25} a^{5} + \frac{2}{5} a^{4} - \frac{12}{25} a^{3} + \frac{8}{25} a^{2} - \frac{2}{5} a$, $\frac{1}{25} a^{10} - \frac{2}{25} a^{6} + \frac{1}{25} a^{2}$, $\frac{1}{125} a^{11} - \frac{2}{125} a^{10} + \frac{2}{125} a^{9} - \frac{2}{125} a^{8} + \frac{1}{25} a^{7} + \frac{12}{125} a^{6} - \frac{7}{125} a^{5} - \frac{18}{125} a^{4} + \frac{14}{125} a^{3} - \frac{6}{25} a^{2} + \frac{1}{5} a$, $\frac{1}{125} a^{12} - \frac{2}{125} a^{10} + \frac{2}{125} a^{9} + \frac{1}{125} a^{8} - \frac{3}{125} a^{7} - \frac{8}{125} a^{6} - \frac{7}{125} a^{5} - \frac{22}{125} a^{4} + \frac{23}{125} a^{3} - \frac{2}{25} a^{2} + \frac{1}{5} a$, $\frac{1}{9985002188278382947517531289125009134023322638581875} a^{13} + \frac{14858373249209470163703880661208252179721492214857}{9985002188278382947517531289125009134023322638581875} a^{12} + \frac{6611277322114336247574043091046100462678420348329}{9985002188278382947517531289125009134023322638581875} a^{11} - \frac{124427653468457171958387862840312760128847092696314}{9985002188278382947517531289125009134023322638581875} a^{10} + \frac{112249437929493739939273806090234417459178215688927}{9985002188278382947517531289125009134023322638581875} a^{9} + \frac{173370436419954755688988893881092907398003790541912}{9985002188278382947517531289125009134023322638581875} a^{8} + \frac{248435578098041201669488726892845867308140157507321}{9985002188278382947517531289125009134023322638581875} a^{7} + \frac{278444869360983596103149865127194641750892311139674}{9985002188278382947517531289125009134023322638581875} a^{6} + \frac{418888713640955784261115025116954179232065104903287}{9985002188278382947517531289125009134023322638581875} a^{5} - \frac{3995322028228957546927376385431668650522063267193234}{9985002188278382947517531289125009134023322638581875} a^{4} - \frac{4384479898235781146726691267178183407732418899937}{1997000437655676589503506257825001826804664527716375} a^{3} + \frac{58613670074694743108955524877240138639493607874267}{399400087531135317900701251565000365360932905543275} a^{2} + \frac{16418482236695228271005913938931976854968377631571}{79880017506227063580140250313000073072186581108655} a + \frac{3829007721096357477737208127590844187818739758361}{15976003501245412716028050062600014614437316221731}$

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

Class group and class number

$C_{2}\times C_{1046982}$, which has order $2093964$ (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:  \( 315114.6966253571 \) (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

\(\Q(\sqrt{-1491}) \), 7.7.128100283921.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.1.0.1}{1} }^{14}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$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}$
$71$71.14.13.1$x^{14} - 71$$14$$1$$13$$C_{14}$$[\ ]_{14}$