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Properties

Label 14.0.10711950282...8411.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 113^{13}$
Root discriminant $139.63$
Ramified primes $3, 113$
Class number $3408$ (GRH)
Class group $[2, 2, 2, 426]$ (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![3805387, 398747, 4546573, 369519, 1495378, 6877, 145654, -34756, 1220, -5925, 908, -82, 61, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 61*x^12 - 82*x^11 + 908*x^10 - 5925*x^9 + 1220*x^8 - 34756*x^7 + 145654*x^6 + 6877*x^5 + 1495378*x^4 + 369519*x^3 + 4546573*x^2 + 398747*x + 3805387)
 
gp: K = bnfinit(x^14 - x^13 + 61*x^12 - 82*x^11 + 908*x^10 - 5925*x^9 + 1220*x^8 - 34756*x^7 + 145654*x^6 + 6877*x^5 + 1495378*x^4 + 369519*x^3 + 4546573*x^2 + 398747*x + 3805387, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 61 x^{12} - 82 x^{11} + 908 x^{10} - 5925 x^{9} + 1220 x^{8} - 34756 x^{7} + 145654 x^{6} + 6877 x^{5} + 1495378 x^{4} + 369519 x^{3} + 4546573 x^{2} + 398747 x + 3805387 \)

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:  \(-1071195028273471735324517808411=-\,3^{7}\cdot 113^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $139.63$
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:  $3, 113$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(339=3\cdot 113\)
Dirichlet character group:    $\lbrace$$\chi_{339}(256,·)$, $\chi_{339}(1,·)$, $\chi_{339}(290,·)$, $\chi_{339}(323,·)$, $\chi_{339}(230,·)$, $\chi_{339}(233,·)$, $\chi_{339}(106,·)$, $\chi_{339}(109,·)$, $\chi_{339}(16,·)$, $\chi_{339}(49,·)$, $\chi_{339}(338,·)$, $\chi_{339}(83,·)$, $\chi_{339}(311,·)$, $\chi_{339}(28,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{71} a^{12} + \frac{11}{71} a^{11} - \frac{2}{71} a^{10} + \frac{21}{71} a^{9} - \frac{12}{71} a^{8} - \frac{11}{71} a^{7} + \frac{20}{71} a^{6} + \frac{5}{71} a^{5} + \frac{27}{71} a^{4} - \frac{22}{71} a^{3} - \frac{15}{71} a^{2} + \frac{27}{71} a$, $\frac{1}{140733780170109963062488712475575041052377} a^{13} + \frac{33080847150597225095719611014361428}{666984740142701246741652665761019151907} a^{12} + \frac{64529080232104546356996025742244208978658}{140733780170109963062488712475575041052377} a^{11} - \frac{20645814583086772188115008326979335738363}{140733780170109963062488712475575041052377} a^{10} - \frac{41141998040968170330175409117171905204813}{140733780170109963062488712475575041052377} a^{9} - \frac{23845319405009794419940913442771117891242}{140733780170109963062488712475575041052377} a^{8} - \frac{60584155340997708413199195123866627558453}{140733780170109963062488712475575041052377} a^{7} - \frac{51197567114576300770863594617858574891242}{140733780170109963062488712475575041052377} a^{6} - \frac{1264205100318284855635560800513703477302}{140733780170109963062488712475575041052377} a^{5} - \frac{10225681772208885587279707287051450473387}{140733780170109963062488712475575041052377} a^{4} + \frac{68398998805901664663415840424670796530700}{140733780170109963062488712475575041052377} a^{3} + \frac{23100013310446669007765632278793338188458}{140733780170109963062488712475575041052377} a^{2} + \frac{51010176257162596438175819353744239202655}{140733780170109963062488712475575041052377} a - \frac{516118046863072955529352865010522950171}{1982165917888872719189981865853169592287}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{426}$, which has order $3408$ (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

\(\Q(\sqrt{-339}) \), 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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$3$3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$113$113.14.13.11$x^{14} + 247131$$14$$1$$13$$C_{14}$$[\ ]_{14}$