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Properties

Label 14.0.60452572724...9375.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 5^{7}\cdot 29^{12}$
Root discriminant $69.43$
Ramified primes $3, 5, 29$
Class number $1808$ (GRH)
Class group $[2, 2, 2, 226]$ (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![499381, -231593, 363084, -143071, 123031, -42720, 22552, -6347, 2438, -514, 150, -19, 13, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + 13*x^12 - 19*x^11 + 150*x^10 - 514*x^9 + 2438*x^8 - 6347*x^7 + 22552*x^6 - 42720*x^5 + 123031*x^4 - 143071*x^3 + 363084*x^2 - 231593*x + 499381)
 
gp: K = bnfinit(x^14 - 5*x^13 + 13*x^12 - 19*x^11 + 150*x^10 - 514*x^9 + 2438*x^8 - 6347*x^7 + 22552*x^6 - 42720*x^5 + 123031*x^4 - 143071*x^3 + 363084*x^2 - 231593*x + 499381, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} + 13 x^{12} - 19 x^{11} + 150 x^{10} - 514 x^{9} + 2438 x^{8} - 6347 x^{7} + 22552 x^{6} - 42720 x^{5} + 123031 x^{4} - 143071 x^{3} + 363084 x^{2} - 231593 x + 499381 \)

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:  \(-60452572724246936927109375=-\,3^{7}\cdot 5^{7}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.43$
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:  $3, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(435=3\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{435}(256,·)$, $\chi_{435}(1,·)$, $\chi_{435}(226,·)$, $\chi_{435}(136,·)$, $\chi_{435}(74,·)$, $\chi_{435}(194,·)$, $\chi_{435}(239,·)$, $\chi_{435}(16,·)$, $\chi_{435}(181,·)$, $\chi_{435}(344,·)$, $\chi_{435}(314,·)$, $\chi_{435}(59,·)$, $\chi_{435}(284,·)$, $\chi_{435}(286,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{17} a^{10} + \frac{2}{17} a^{9} - \frac{1}{17} a^{8} + \frac{5}{17} a^{7} - \frac{2}{17} a^{6} - \frac{3}{17} a^{5} - \frac{1}{17} a^{4} - \frac{8}{17} a^{3} + \frac{6}{17} a^{2} + \frac{4}{17} a - \frac{3}{17}$, $\frac{1}{17} a^{11} - \frac{5}{17} a^{9} + \frac{7}{17} a^{8} + \frac{5}{17} a^{7} + \frac{1}{17} a^{6} + \frac{5}{17} a^{5} - \frac{6}{17} a^{4} + \frac{5}{17} a^{3} - \frac{8}{17} a^{2} + \frac{6}{17} a + \frac{6}{17}$, $\frac{1}{289} a^{12} + \frac{7}{289} a^{11} + \frac{2}{289} a^{10} - \frac{82}{289} a^{9} + \frac{132}{289} a^{8} - \frac{99}{289} a^{7} - \frac{19}{289} a^{6} - \frac{60}{289} a^{5} - \frac{61}{289} a^{4} + \frac{107}{289} a^{3} + \frac{60}{289} a^{2} + \frac{76}{289} a - \frac{64}{289}$, $\frac{1}{49954749402653209833685051} a^{13} + \frac{20474162711788568925686}{49954749402653209833685051} a^{12} + \frac{1225849218386583202897417}{49954749402653209833685051} a^{11} + \frac{952048006077019298479002}{49954749402653209833685051} a^{10} + \frac{17680312965927867478938792}{49954749402653209833685051} a^{9} - \frac{1526592645538728788182583}{49954749402653209833685051} a^{8} - \frac{4966238526598083748174929}{49954749402653209833685051} a^{7} - \frac{1367605032513507511513955}{2938514670744306460805003} a^{6} - \frac{13285375263035467735229979}{49954749402653209833685051} a^{5} - \frac{127883435499842342574017}{49954749402653209833685051} a^{4} + \frac{17437663692467984491694208}{49954749402653209833685051} a^{3} + \frac{12355859171554516684875995}{49954749402653209833685051} a^{2} + \frac{689628757311151956732043}{49954749402653209833685051} a - \frac{3400837463200400316087036}{49954749402653209833685051}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{226}$, which has order $1808$ (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:  \( 6020.985100147561 \) (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

\(\Q(\sqrt{-15}) \), 7.7.594823321.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ R R ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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}$$[\ ]_{2}^{7}$
$5$5.14.7.2$x^{14} - 15625 x^{2} + 156250$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$