↑

Properties

Label 14.0.13760207184...6523.6
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 29^{13}$
Root discriminant $736.28$
Ramified primes $7, 29$
Class number $2124668$ (GRH)
Class group $[13, 163436]$ (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![5899794618083, -2109103641127, 188327791736, -26703542026, 5834381182, 756770602, -133193172, -18378373, 1069810, 125454, 24360, -2233, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2233*x^11 + 24360*x^10 + 125454*x^9 + 1069810*x^8 - 18378373*x^7 - 133193172*x^6 + 756770602*x^5 + 5834381182*x^4 - 26703542026*x^3 + 188327791736*x^2 - 2109103641127*x + 5899794618083)
 
gp: K = bnfinit(x^14 - 2233*x^11 + 24360*x^10 + 125454*x^9 + 1069810*x^8 - 18378373*x^7 - 133193172*x^6 + 756770602*x^5 + 5834381182*x^4 - 26703542026*x^3 + 188327791736*x^2 - 2109103641127*x + 5899794618083, 1)
 

Normalized defining polynomial

\( x^{14} - 2233 x^{11} + 24360 x^{10} + 125454 x^{9} + 1069810 x^{8} - 18378373 x^{7} - 133193172 x^{6} + 756770602 x^{5} + 5834381182 x^{4} - 26703542026 x^{3} + 188327791736 x^{2} - 2109103641127 x + 5899794618083 \)

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:  \(-13760207184971837367389621368916758066523=-\,7^{25}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $736.28$
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:  $7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1421=7^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1421}(1,·)$, $\chi_{1421}(547,·)$, $\chi_{1421}(806,·)$, $\chi_{1421}(615,·)$, $\chi_{1421}(874,·)$, $\chi_{1421}(1420,·)$, $\chi_{1421}(622,·)$, $\chi_{1421}(239,·)$, $\chi_{1421}(372,·)$, $\chi_{1421}(281,·)$, $\chi_{1421}(1140,·)$, $\chi_{1421}(799,·)$, $\chi_{1421}(1182,·)$, $\chi_{1421}(1049,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{23} a^{9} - \frac{8}{23} a^{8} + \frac{1}{23} a^{7} - \frac{4}{23} a^{6} + \frac{2}{23} a^{5} - \frac{6}{23} a^{4} + \frac{8}{23} a^{3} - \frac{9}{23} a^{2} - \frac{9}{23} a + \frac{7}{23}$, $\frac{1}{23} a^{10} + \frac{6}{23} a^{8} + \frac{4}{23} a^{7} - \frac{7}{23} a^{6} + \frac{10}{23} a^{5} + \frac{6}{23} a^{4} + \frac{9}{23} a^{3} + \frac{11}{23} a^{2} + \frac{4}{23} a + \frac{10}{23}$, $\frac{1}{23} a^{11} + \frac{6}{23} a^{8} + \frac{10}{23} a^{7} + \frac{11}{23} a^{6} - \frac{6}{23} a^{5} - \frac{1}{23} a^{4} + \frac{9}{23} a^{3} - \frac{11}{23} a^{2} - \frac{5}{23} a + \frac{4}{23}$, $\frac{1}{3473} a^{12} - \frac{18}{3473} a^{11} - \frac{74}{3473} a^{10} + \frac{72}{3473} a^{9} - \frac{334}{3473} a^{8} + \frac{130}{3473} a^{7} + \frac{556}{3473} a^{6} + \frac{603}{3473} a^{5} + \frac{1004}{3473} a^{4} + \frac{724}{3473} a^{3} - \frac{1606}{3473} a^{2} + \frac{147}{3473} a - \frac{258}{3473}$, $\frac{1}{1255815893293075750397942618089950589221778358826543404183108874389} a^{13} + \frac{58323077973116125693616361354063653451631680299207966042706074}{1255815893293075750397942618089950589221778358826543404183108874389} a^{12} - \frac{6911863697946168835392464631760843232830369627750267764625848423}{1255815893293075750397942618089950589221778358826543404183108874389} a^{11} + \frac{23693520865991039430127075551092430622848278922973184544196320009}{1255815893293075750397942618089950589221778358826543404183108874389} a^{10} + \frac{13013581595780680446917129388713896440076570335634336130796918059}{1255815893293075750397942618089950589221778358826543404183108874389} a^{9} - \frac{238194282661242944875981754928719708106430193354762864181258849437}{1255815893293075750397942618089950589221778358826543404183108874389} a^{8} + \frac{440081459787081630804239813576156573995561091940856250189738114495}{1255815893293075750397942618089950589221778358826543404183108874389} a^{7} + \frac{416048383000957208771870199232145999826048974325971084360220232076}{1255815893293075750397942618089950589221778358826543404183108874389} a^{6} + \frac{570132389036693530073050194679990550519894597826111482383643616199}{1255815893293075750397942618089950589221778358826543404183108874389} a^{5} + \frac{361317375997747661179284721105641186409312205334255562180219478285}{1255815893293075750397942618089950589221778358826543404183108874389} a^{4} - \frac{615829241128155768787047435919186367261166264445818345077504828104}{1255815893293075750397942618089950589221778358826543404183108874389} a^{3} - \frac{172051391350817591794560139250950490207866537742795948384751503769}{1255815893293075750397942618089950589221778358826543404183108874389} a^{2} - \frac{431220638766052787389590596599363508639212657788264621812709518250}{1255815893293075750397942618089950589221778358826543404183108874389} a + \frac{147368742173318308553022394700074806796056404612677702364832684431}{1255815893293075750397942618089950589221778358826543404183108874389}$

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

Class group and class number

$C_{13}\times C_{163436}$, which has order $2124668$ (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:  \( 78882035.29112078 \) (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

\(\Q(\sqrt{-203}) \), 7.7.8233120419813614521.4

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{14}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$7$7.14.25.89$x^{14} - 14 x^{13} + 21 x^{12} + 14 x^{7} - 21$$14$$1$$25$$C_{14}$$[2]_{2}$
$29$29.14.13.4$x^{14} - 1856$$14$$1$$13$$C_{14}$$[\ ]_{14}$