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Properties

Label 14.0.78294573987...1792.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 7^{24}\cdot 11^{7}$
Root discriminant $263.62$
Ramified primes $2, 7, 11$
Class number $5904502$ (GRH)
Class group $[5904502]$ (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![11696930513, 1257735836, 2128994980, 148149106, 178583580, 7273378, 9029657, 151440, 303989, 182, 7091, -42, 112, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 112*x^12 - 42*x^11 + 7091*x^10 + 182*x^9 + 303989*x^8 + 151440*x^7 + 9029657*x^6 + 7273378*x^5 + 178583580*x^4 + 148149106*x^3 + 2128994980*x^2 + 1257735836*x + 11696930513)
 
gp: K = bnfinit(x^14 + 112*x^12 - 42*x^11 + 7091*x^10 + 182*x^9 + 303989*x^8 + 151440*x^7 + 9029657*x^6 + 7273378*x^5 + 178583580*x^4 + 148149106*x^3 + 2128994980*x^2 + 1257735836*x + 11696930513, 1)
 

Normalized defining polynomial

\( x^{14} + 112 x^{12} - 42 x^{11} + 7091 x^{10} + 182 x^{9} + 303989 x^{8} + 151440 x^{7} + 9029657 x^{6} + 7273378 x^{5} + 178583580 x^{4} + 148149106 x^{3} + 2128994980 x^{2} + 1257735836 x + 11696930513 \)

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:  \(-7829457398773661133521078601121792=-\,2^{21}\cdot 7^{24}\cdot 11^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $263.62$
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:  $2, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4312=2^{3}\cdot 7^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{4312}(1,·)$, $\chi_{4312}(617,·)$, $\chi_{4312}(197,·)$, $\chi_{4312}(2465,·)$, $\chi_{4312}(3081,·)$, $\chi_{4312}(813,·)$, $\chi_{4312}(2661,·)$, $\chi_{4312}(3277,·)$, $\chi_{4312}(1233,·)$, $\chi_{4312}(3893,·)$, $\chi_{4312}(3697,·)$, $\chi_{4312}(1849,·)$, $\chi_{4312}(2045,·)$, $\chi_{4312}(1429,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19} a^{9} - \frac{5}{19} a^{8} - \frac{5}{19} a^{7} + \frac{1}{19} a^{6} - \frac{3}{19} a^{5} + \frac{9}{19} a^{4} + \frac{1}{19} a^{3} - \frac{4}{19} a^{2} + \frac{4}{19} a - \frac{3}{19}$, $\frac{1}{19} a^{10} + \frac{8}{19} a^{8} - \frac{5}{19} a^{7} + \frac{2}{19} a^{6} - \frac{6}{19} a^{5} + \frac{8}{19} a^{4} + \frac{1}{19} a^{3} + \frac{3}{19} a^{2} - \frac{2}{19} a + \frac{4}{19}$, $\frac{1}{589} a^{11} + \frac{3}{589} a^{10} - \frac{7}{589} a^{9} - \frac{20}{589} a^{8} - \frac{185}{589} a^{7} + \frac{4}{589} a^{6} + \frac{244}{589} a^{5} + \frac{156}{589} a^{4} - \frac{218}{589} a^{3} + \frac{86}{589} a^{2} - \frac{252}{589} a - \frac{3}{31}$, $\frac{1}{589} a^{12} + \frac{15}{589} a^{10} + \frac{1}{589} a^{9} + \frac{123}{589} a^{8} - \frac{185}{589} a^{7} + \frac{294}{589} a^{6} - \frac{173}{589} a^{5} + \frac{151}{589} a^{4} + \frac{182}{589} a^{3} + \frac{172}{589} a^{2} + \frac{48}{589} a - \frac{294}{589}$, $\frac{1}{3404376010067425829404949914019932090793} a^{13} + \frac{56581278838042306864440775873840952}{109818580969916962238869352065159099703} a^{12} + \frac{1674959448767777090521648965452997203}{3404376010067425829404949914019932090793} a^{11} - \frac{62745657300979758042567900369866796208}{3404376010067425829404949914019932090793} a^{10} - \frac{71963279492274966694758385989851724281}{3404376010067425829404949914019932090793} a^{9} + \frac{256163035833464011826496000450153132587}{3404376010067425829404949914019932090793} a^{8} - \frac{817928901734206475862761496364708956166}{3404376010067425829404949914019932090793} a^{7} - \frac{818380466974740990594115645114424128958}{3404376010067425829404949914019932090793} a^{6} - \frac{7718895357012075899285751542462519670}{109818580969916962238869352065159099703} a^{5} + \frac{928257684867089375945782627335895235381}{3404376010067425829404949914019932090793} a^{4} - \frac{746192719416876422505530882775767776947}{3404376010067425829404949914019932090793} a^{3} + \frac{1573426293138942132453889699460663771603}{3404376010067425829404949914019932090793} a^{2} - \frac{1398317551437647566458562306741198424505}{3404376010067425829404949914019932090793} a - \frac{1137931007941226163372809316021146931197}{3404376010067425829404949914019932090793}$

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

Class group and class number

$C_{5904502}$, which has order $5904502$ (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:  \( 35256.68973693789 \) (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

\(\Q(\sqrt{-22}) \), 7.7.13841287201.1

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 R ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R R ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$2$2.14.21.33$x^{14} + 4 x^{13} + 4 x^{12} + 4 x^{11} - 3 x^{10} + 4 x^{9} - 2 x^{7} - x^{6} - 2 x^{5} + 2 x^{4} - 2 x^{3} + 3 x^{2} + 2 x + 1$$2$$7$$21$$C_{14}$$[3]^{7}$
$7$7.14.24.53$x^{14} + 931 x^{13} + 2310 x^{12} + 903 x^{11} + 392 x^{10} + 2198 x^{9} + 2296 x^{8} + 1485 x^{7} + 637 x^{6} + 1295 x^{5} + 2303 x^{4} + 1449 x^{3} + 1316 x^{2} + 2219 x + 2383$$7$$2$$24$$C_{14}$$[2]^{2}$
$11$11.14.7.1$x^{14} - 2662 x^{8} + 1771561 x^{2} - 311794736$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$