↑

Properties

Label 14.0.83018085220...4319.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,239^{13}$
Root discriminant $161.63$
Ramified prime $239$
Class number $960$ (GRH)
Class group $[4, 4, 60]$ (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![921317, 1090787, 452768, -5712, -37491, -33400, 92428, 10839, -4853, -5214, 170, 373, 9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 9*x^12 + 373*x^11 + 170*x^10 - 5214*x^9 - 4853*x^8 + 10839*x^7 + 92428*x^6 - 33400*x^5 - 37491*x^4 - 5712*x^3 + 452768*x^2 + 1090787*x + 921317)
 
gp: K = bnfinit(x^14 - x^13 + 9*x^12 + 373*x^11 + 170*x^10 - 5214*x^9 - 4853*x^8 + 10839*x^7 + 92428*x^6 - 33400*x^5 - 37491*x^4 - 5712*x^3 + 452768*x^2 + 1090787*x + 921317, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 9 x^{12} + 373 x^{11} + 170 x^{10} - 5214 x^{9} - 4853 x^{8} + 10839 x^{7} + 92428 x^{6} - 33400 x^{5} - 37491 x^{4} - 5712 x^{3} + 452768 x^{2} + 1090787 x + 921317 \)

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:  \(-8301808522036075428899283574319=-\,239^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $161.63$
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:  $239$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(239\)
Dirichlet character group:    $\lbrace$$\chi_{239}(1,·)$, $\chi_{239}(98,·)$, $\chi_{239}(195,·)$, $\chi_{239}(100,·)$, $\chi_{239}(229,·)$, $\chi_{239}(38,·)$, $\chi_{239}(201,·)$, $\chi_{239}(10,·)$, $\chi_{239}(139,·)$, $\chi_{239}(44,·)$, $\chi_{239}(141,·)$, $\chi_{239}(238,·)$, $\chi_{239}(215,·)$, $\chi_{239}(24,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{71} a^{11} + \frac{21}{71} a^{10} - \frac{32}{71} a^{9} - \frac{6}{71} a^{8} - \frac{26}{71} a^{7} - \frac{18}{71} a^{6} + \frac{11}{71} a^{5} + \frac{31}{71} a^{4} + \frac{10}{71} a^{3} - \frac{5}{71} a^{2} + \frac{34}{71} a + \frac{2}{71}$, $\frac{1}{71} a^{12} + \frac{24}{71} a^{10} + \frac{27}{71} a^{9} + \frac{29}{71} a^{8} + \frac{31}{71} a^{7} + \frac{34}{71} a^{6} + \frac{13}{71} a^{5} - \frac{2}{71} a^{4} - \frac{2}{71} a^{3} - \frac{3}{71} a^{2} - \frac{2}{71} a + \frac{29}{71}$, $\frac{1}{92493629435290649256747771267915522971} a^{13} - \frac{216604390813316722956660176229547269}{92493629435290649256747771267915522971} a^{12} + \frac{287213086315073601111920564737126404}{92493629435290649256747771267915522971} a^{11} + \frac{12148207894845342199503325229871228886}{92493629435290649256747771267915522971} a^{10} + \frac{17787455536351581507584513344532479492}{92493629435290649256747771267915522971} a^{9} - \frac{33167752558412032937245605086791016418}{92493629435290649256747771267915522971} a^{8} + \frac{16228121840339071043127354251400733004}{92493629435290649256747771267915522971} a^{7} - \frac{7699724994469494677135895755477276996}{92493629435290649256747771267915522971} a^{6} - \frac{1757774490730798423329599963760383846}{92493629435290649256747771267915522971} a^{5} + \frac{36796411768628278929183519846316049616}{92493629435290649256747771267915522971} a^{4} + \frac{202713037234894862078225455981505197}{1380501931870009690399220466685306313} a^{3} - \frac{11362923746596263591820703738068974212}{92493629435290649256747771267915522971} a^{2} + \frac{11578490265025168501499827512761876584}{92493629435290649256747771267915522971} a + \frac{29951726136789392453440840945649998}{1380501931870009690399220466685306313}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{60}$, which has order $960$ (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:  \( 3022802.06733 \) (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

\(\Q(\sqrt{-239}) \), 7.7.186374892382561.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
239Data not computed