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Properties

Label 14.0.32733449455...9375.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 5^{7}\cdot 7^{24}$
Root discriminant $108.84$
Ramified primes $3, 5, 7$
Class number $12238$ (GRH)
Class group $[12238]$ (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![2290621, 807429, 797146, 13965, 104601, -672, 27482, -1217, 966, -700, 322, 7, 7, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 7*x^12 + 7*x^11 + 322*x^10 - 700*x^9 + 966*x^8 - 1217*x^7 + 27482*x^6 - 672*x^5 + 104601*x^4 + 13965*x^3 + 797146*x^2 + 807429*x + 2290621)
 
gp: K = bnfinit(x^14 - 7*x^13 + 7*x^12 + 7*x^11 + 322*x^10 - 700*x^9 + 966*x^8 - 1217*x^7 + 27482*x^6 - 672*x^5 + 104601*x^4 + 13965*x^3 + 797146*x^2 + 807429*x + 2290621, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 7 x^{12} + 7 x^{11} + 322 x^{10} - 700 x^{9} + 966 x^{8} - 1217 x^{7} + 27482 x^{6} - 672 x^{5} + 104601 x^{4} + 13965 x^{3} + 797146 x^{2} + 807429 x + 2290621 \)

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:  \(-32733449455413964710545859375=-\,3^{7}\cdot 5^{7}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $108.84$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(735=3\cdot 5\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{735}(1,·)$, $\chi_{735}(421,·)$, $\chi_{735}(134,·)$, $\chi_{735}(449,·)$, $\chi_{735}(554,·)$, $\chi_{735}(29,·)$, $\chi_{735}(526,·)$, $\chi_{735}(239,·)$, $\chi_{735}(659,·)$, $\chi_{735}(631,·)$, $\chi_{735}(344,·)$, $\chi_{735}(316,·)$, $\chi_{735}(106,·)$, $\chi_{735}(211,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19} a^{10} - \frac{9}{19} a^{9} - \frac{4}{19} a^{8} + \frac{2}{19} a^{7} - \frac{5}{19} a^{6} + \frac{7}{19} a^{5} + \frac{9}{19} a^{4} - \frac{7}{19} a^{3} - \frac{1}{19} a^{2} + \frac{7}{19} a$, $\frac{1}{589} a^{11} + \frac{12}{589} a^{10} + \frac{244}{589} a^{9} - \frac{139}{589} a^{8} - \frac{58}{589} a^{7} + \frac{263}{589} a^{6} + \frac{175}{589} a^{5} - \frac{236}{589} a^{4} - \frac{262}{589} a^{3} + \frac{43}{589} a^{2} - \frac{43}{589} a$, $\frac{1}{589} a^{12} + \frac{7}{589} a^{10} + \frac{126}{589} a^{9} + \frac{215}{589} a^{8} + \frac{184}{589} a^{7} - \frac{160}{589} a^{6} - \frac{42}{589} a^{5} - \frac{34}{589} a^{4} - \frac{15}{31} a^{3} + \frac{123}{589} a^{2} - \frac{135}{589} a$, $\frac{1}{24427730328100378479544387669} a^{13} - \frac{2261054786297483705410806}{24427730328100378479544387669} a^{12} + \frac{19468078206963442525222018}{24427730328100378479544387669} a^{11} + \frac{119389730203826785569016673}{24427730328100378479544387669} a^{10} + \frac{734640333445762470112818510}{24427730328100378479544387669} a^{9} - \frac{4090899366795463850027322550}{24427730328100378479544387669} a^{8} - \frac{3606418266839087625509834515}{24427730328100378479544387669} a^{7} + \frac{185524120095054316098282795}{24427730328100378479544387669} a^{6} + \frac{6824295983184440222649681821}{24427730328100378479544387669} a^{5} - \frac{8662002307866148147687362315}{24427730328100378479544387669} a^{4} + \frac{10587230119354321780649206546}{24427730328100378479544387669} a^{3} - \frac{10025860669921350594864103746}{24427730328100378479544387669} a^{2} - \frac{6745702226473521966721722918}{24427730328100378479544387669} a + \frac{18009181056712155289446797}{41473226363498095890567721}$

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

Class group and class number

$C_{12238}$, which has order $12238$ (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

\(\Q(\sqrt{-15}) \), 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ R R R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$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}$
$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}$