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Properties

Label 14.0.74721141340...7867.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 197^{12}$
Root discriminant $160.42$
Ramified primes $3, 197$
Class number $6728$ (GRH)
Class group $[2, 2, 1682]$ (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![1352569, -1666579, 6691533, 2579356, 17583457, -4851594, 2559802, -374872, 153467, -19520, 5925, -350, 85, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 85*x^12 - 350*x^11 + 5925*x^10 - 19520*x^9 + 153467*x^8 - 374872*x^7 + 2559802*x^6 - 4851594*x^5 + 17583457*x^4 + 2579356*x^3 + 6691533*x^2 - 1666579*x + 1352569)
 
gp: K = bnfinit(x^14 - x^13 + 85*x^12 - 350*x^11 + 5925*x^10 - 19520*x^9 + 153467*x^8 - 374872*x^7 + 2559802*x^6 - 4851594*x^5 + 17583457*x^4 + 2579356*x^3 + 6691533*x^2 - 1666579*x + 1352569, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 85 x^{12} - 350 x^{11} + 5925 x^{10} - 19520 x^{9} + 153467 x^{8} - 374872 x^{7} + 2559802 x^{6} - 4851594 x^{5} + 17583457 x^{4} + 2579356 x^{3} + 6691533 x^{2} - 1666579 x + 1352569 \)

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:  \(-7472114134063096576756423407867=-\,3^{7}\cdot 197^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $160.42$
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:  $3, 197$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(591=3\cdot 197\)
Dirichlet character group:    $\lbrace$$\chi_{591}(1,·)$, $\chi_{591}(164,·)$, $\chi_{591}(311,·)$, $\chi_{591}(104,·)$, $\chi_{591}(361,·)$, $\chi_{591}(395,·)$, $\chi_{591}(301,·)$, $\chi_{591}(430,·)$, $\chi_{591}(178,·)$, $\chi_{591}(233,·)$, $\chi_{591}(388,·)$, $\chi_{591}(572,·)$, $\chi_{591}(508,·)$, $\chi_{591}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{28070353} a^{12} + \frac{8218419}{28070353} a^{11} - \frac{11173109}{28070353} a^{10} - \frac{1497578}{28070353} a^{9} + \frac{24381}{28070353} a^{8} + \frac{10418398}{28070353} a^{7} + \frac{11501117}{28070353} a^{6} + \frac{12117543}{28070353} a^{5} + \frac{3491565}{28070353} a^{4} + \frac{9336567}{28070353} a^{3} - \frac{10445278}{28070353} a^{2} + \frac{4983298}{28070353} a - \frac{10645837}{28070353}$, $\frac{1}{949309559816401131074207340684842683397} a^{13} + \frac{9121712470144868215267119206500}{949309559816401131074207340684842683397} a^{12} - \frac{164722489810927287724951842564378957073}{949309559816401131074207340684842683397} a^{11} - \frac{5964449842958738796447420549432062977}{49963661042968480582853017930781193863} a^{10} + \frac{39242735769325472045452156086334001235}{949309559816401131074207340684842683397} a^{9} + \frac{383015540551735227511545984038785584814}{949309559816401131074207340684842683397} a^{8} - \frac{175941439156458938807195323913514744063}{949309559816401131074207340684842683397} a^{7} + \frac{45921673875988490864201079938780682505}{949309559816401131074207340684842683397} a^{6} - \frac{88705309365535038009802134643447999511}{949309559816401131074207340684842683397} a^{5} + \frac{2592620378375665512660132525692949552}{49963661042968480582853017930781193863} a^{4} - \frac{440432932604208956160613814898985364010}{949309559816401131074207340684842683397} a^{3} + \frac{459950791054954309390437074196780746722}{949309559816401131074207340684842683397} a^{2} - \frac{63360841354500067582086103147133995576}{949309559816401131074207340684842683397} a + \frac{277840124062356716064444251773482562}{816259294769046544345836062497715119}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{1682}$, which has order $6728$ (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:  \( -\frac{130972546947799148166776610}{642559843708115159449898598461231} a^{13} + \frac{36842029464140620802254010}{642559843708115159449898598461231} a^{12} - \frac{10999449771664585050440540230}{642559843708115159449898598461231} a^{11} + \frac{1989663145850003295574544625}{33818939142532376813152557813749} a^{10} - \frac{739876571375117560645494280725}{642559843708115159449898598461231} a^{9} + \frac{1985573723705941248870299748247}{642559843708115159449898598461231} a^{8} - \frac{18042080309213789690594351884870}{642559843708115159449898598461231} a^{7} + \frac{33946242330242288858091431262345}{642559843708115159449898598461231} a^{6} - \frac{294729385338472773817743752983526}{642559843708115159449898598461231} a^{5} + \frac{20109763778129236267548724979810}{33818939142532376813152557813749} a^{4} - \frac{1760719407598003247984362558484245}{642559843708115159449898598461231} a^{3} - \frac{2171698275825359397813969946531897}{642559843708115159449898598461231} a^{2} - \frac{653906744824477310268113402497305}{642559843708115159449898598461231} a + \frac{135268528783275862761944710505}{552502015226238314230351331437} \) (order $6$)
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:  \( 1553055.199048291 \) (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

\(\Q(\sqrt{-3}) \), 7.7.58451728309129.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$197$197.14.12.1$x^{14} + 16351 x^{7} + 84875283$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$