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Properties

Label 14.0.35696661575...9383.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{7}\cdot 113^{12}$
Root discriminant $152.17$
Ramified primes $7, 113$
Class number $6328$ (GRH)
Class group $[2, 2, 1582]$ (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![1305077, -1429473, 2003814, -1091251, 670781, -117192, 48006, -19951, 27880, -13306, 1036, 489, -73, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 - 73*x^12 + 489*x^11 + 1036*x^10 - 13306*x^9 + 27880*x^8 - 19951*x^7 + 48006*x^6 - 117192*x^5 + 670781*x^4 - 1091251*x^3 + 2003814*x^2 - 1429473*x + 1305077)
 
gp: K = bnfinit(x^14 - 5*x^13 - 73*x^12 + 489*x^11 + 1036*x^10 - 13306*x^9 + 27880*x^8 - 19951*x^7 + 48006*x^6 - 117192*x^5 + 670781*x^4 - 1091251*x^3 + 2003814*x^2 - 1429473*x + 1305077, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} - 73 x^{12} + 489 x^{11} + 1036 x^{10} - 13306 x^{9} + 27880 x^{8} - 19951 x^{7} + 48006 x^{6} - 117192 x^{5} + 670781 x^{4} - 1091251 x^{3} + 2003814 x^{2} - 1429473 x + 1305077 \)

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:  \(-3569666157501162271525463699383=-\,7^{7}\cdot 113^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $152.17$
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:  $7, 113$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(791=7\cdot 113\)
Dirichlet character group:    $\lbrace$$\chi_{791}(1,·)$, $\chi_{791}(162,·)$, $\chi_{791}(708,·)$, $\chi_{791}(566,·)$, $\chi_{791}(106,·)$, $\chi_{791}(482,·)$, $\chi_{791}(141,·)$, $\chi_{791}(561,·)$, $\chi_{791}(468,·)$, $\chi_{791}(694,·)$, $\chi_{791}(727,·)$, $\chi_{791}(706,·)$, $\chi_{791}(335,·)$, $\chi_{791}(671,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11461} a^{12} - \frac{4666}{11461} a^{11} - \frac{3543}{11461} a^{10} - \frac{59}{11461} a^{9} - \frac{2599}{11461} a^{8} + \frac{2406}{11461} a^{7} + \frac{2683}{11461} a^{6} + \frac{2942}{11461} a^{5} - \frac{1838}{11461} a^{4} + \frac{3944}{11461} a^{3} - \frac{308}{11461} a^{2} + \frac{2414}{11461} a - \frac{3995}{11461}$, $\frac{1}{5933303819862530455651424600953247} a^{13} - \frac{236538123597949589103619352027}{5933303819862530455651424600953247} a^{12} + \frac{1690014005945078755298688153802281}{5933303819862530455651424600953247} a^{11} + \frac{1166376853765355730858080551930483}{5933303819862530455651424600953247} a^{10} - \frac{31936188581959201855543840310857}{81278134518664800762348282204839} a^{9} + \frac{966889306591300584414317478562797}{5933303819862530455651424600953247} a^{8} + \frac{1423605754894854075781067765819757}{5933303819862530455651424600953247} a^{7} - \frac{1482388940937217444032526560014644}{5933303819862530455651424600953247} a^{6} - \frac{630128947832075518625322249690597}{5933303819862530455651424600953247} a^{5} + \frac{1010683412418715568816469912072444}{5933303819862530455651424600953247} a^{4} - \frac{2731206450662148371081049399226277}{5933303819862530455651424600953247} a^{3} - \frac{1637688838367245766924235686894142}{5933303819862530455651424600953247} a^{2} + \frac{2390981518735567203639115061902849}{5933303819862530455651424600953247} a + \frac{2081886498938846763246402595395183}{5933303819862530455651424600953247}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{1582}$, which has order $6328$ (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:  \( 222748.97284811488 \) (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

\(\Q(\sqrt{-7}) \), 7.7.2081951752609.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$7$7.14.7.1$x^{14} - 117649 x^{2} + 1647086$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$113$113.7.6.1$x^{7} - 113$$7$$1$$6$$C_7$$[\ ]_{7}$
113.7.6.1$x^{7} - 113$$7$$1$$6$$C_7$$[\ ]_{7}$