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Properties

Label 14.0.52075344228...1907.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{24}\cdot 43^{7}$
Root discriminant $184.28$
Ramified primes $7, 43$
Class number $297431$ (GRH)
Class group $[297431]$ (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![304880351, -143330628, 98359275, -32204158, 13681430, -3388448, 1046059, -197585, 50645, -7273, 1624, -203, 56, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 56*x^12 - 203*x^11 + 1624*x^10 - 7273*x^9 + 50645*x^8 - 197585*x^7 + 1046059*x^6 - 3388448*x^5 + 13681430*x^4 - 32204158*x^3 + 98359275*x^2 - 143330628*x + 304880351)
 
gp: K = bnfinit(x^14 - 7*x^13 + 56*x^12 - 203*x^11 + 1624*x^10 - 7273*x^9 + 50645*x^8 - 197585*x^7 + 1046059*x^6 - 3388448*x^5 + 13681430*x^4 - 32204158*x^3 + 98359275*x^2 - 143330628*x + 304880351, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 56 x^{12} - 203 x^{11} + 1624 x^{10} - 7273 x^{9} + 50645 x^{8} - 197585 x^{7} + 1046059 x^{6} - 3388448 x^{5} + 13681430 x^{4} - 32204158 x^{3} + 98359275 x^{2} - 143330628 x + 304880351 \)

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:  \(-52075344228034366913450823351907=-\,7^{24}\cdot 43^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $184.28$
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:  $7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2107=7^{2}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{2107}(1,·)$, $\chi_{2107}(386,·)$, $\chi_{2107}(1891,·)$, $\chi_{2107}(904,·)$, $\chi_{2107}(1289,·)$, $\chi_{2107}(1506,·)$, $\chi_{2107}(302,·)$, $\chi_{2107}(1807,·)$, $\chi_{2107}(85,·)$, $\chi_{2107}(1590,·)$, $\chi_{2107}(603,·)$, $\chi_{2107}(988,·)$, $\chi_{2107}(687,·)$, $\chi_{2107}(1205,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{31} a^{11} + \frac{15}{31} a^{10} + \frac{11}{31} a^{9} - \frac{13}{31} a^{8} - \frac{9}{31} a^{7} - \frac{7}{31} a^{6} - \frac{14}{31} a^{5} + \frac{2}{31} a^{4} + \frac{6}{31} a^{3} + \frac{14}{31} a^{2} + \frac{11}{31} a + \frac{9}{31}$, $\frac{1}{589} a^{12} - \frac{9}{589} a^{11} - \frac{256}{589} a^{10} - \frac{122}{589} a^{9} - \frac{7}{589} a^{8} - \frac{287}{589} a^{7} - \frac{32}{589} a^{6} + \frac{245}{589} a^{5} - \frac{290}{589} a^{4} - \frac{192}{589} a^{3} - \frac{108}{589} a^{2} + \frac{179}{589} a - \frac{154}{589}$, $\frac{1}{74769322456297736075668933789987864577} a^{13} - \frac{50349819085987298682506938470773329}{74769322456297736075668933789987864577} a^{12} - \frac{1111083883687618891135329461463833746}{74769322456297736075668933789987864577} a^{11} - \frac{231312925037604550912947802234688807}{3935227497699880846087838620525677083} a^{10} + \frac{267024112035871324828196874880716510}{2411913627622507615344159154515737567} a^{9} + \frac{188707368621251907168628995555081}{1606871170967693281375189310137067} a^{8} + \frac{16516790727347308833137318214540671997}{74769322456297736075668933789987864577} a^{7} - \frac{2274761854884803989562055060084674382}{74769322456297736075668933789987864577} a^{6} - \frac{26032867730010199422043305956759304472}{74769322456297736075668933789987864577} a^{5} + \frac{2893153681782374737106929198950239340}{74769322456297736075668933789987864577} a^{4} + \frac{10842629390083101141815546618177175747}{74769322456297736075668933789987864577} a^{3} + \frac{19933416525794130810293395501994639152}{74769322456297736075668933789987864577} a^{2} + \frac{18985574346243402211001606727997279611}{74769322456297736075668933789987864577} a + \frac{237660276528324697711491593216428737}{1115960036661160239935357220746087531}$

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

Class group and class number

$C_{297431}$, which has order $297431$ (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

\(\Q(\sqrt{-43}) \), 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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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$7.14.24.53$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}$$[2]^{2}$
$43$43.14.7.2$x^{14} - 12642726098 x^{2} + 2446367499963$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$