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Properties

Label 14.0.69014072539...9936.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 7^{7}\cdot 43^{12}$
Root discriminant $188.02$
Ramified primes $2, 7, 43$
Class number $545024$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 8516]$ (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![770198807, -217576562, 216370161, -42685580, 26185174, -3652314, 1808214, -162296, 80327, -3924, 2514, -62, 63, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 63*x^12 - 62*x^11 + 2514*x^10 - 3924*x^9 + 80327*x^8 - 162296*x^7 + 1808214*x^6 - 3652314*x^5 + 26185174*x^4 - 42685580*x^3 + 216370161*x^2 - 217576562*x + 770198807)
 
gp: K = bnfinit(x^14 - 2*x^13 + 63*x^12 - 62*x^11 + 2514*x^10 - 3924*x^9 + 80327*x^8 - 162296*x^7 + 1808214*x^6 - 3652314*x^5 + 26185174*x^4 - 42685580*x^3 + 216370161*x^2 - 217576562*x + 770198807, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 63 x^{12} - 62 x^{11} + 2514 x^{10} - 3924 x^{9} + 80327 x^{8} - 162296 x^{7} + 1808214 x^{6} - 3652314 x^{5} + 26185174 x^{4} - 42685580 x^{3} + 216370161 x^{2} - 217576562 x + 770198807 \)

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:  \(-69014072539312321109968799399936=-\,2^{21}\cdot 7^{7}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $188.02$
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:  $2, 7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2408=2^{3}\cdot 7\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{2408}(1,·)$, $\chi_{2408}(1301,·)$, $\chi_{2408}(517,·)$, $\chi_{2408}(1569,·)$, $\chi_{2408}(2185,·)$, $\chi_{2408}(1245,·)$, $\chi_{2408}(785,·)$, $\chi_{2408}(2197,·)$, $\chi_{2408}(1177,·)$, $\chi_{2408}(1681,·)$, $\chi_{2408}(729,·)$, $\chi_{2408}(293,·)$, $\chi_{2408}(1693,·)$, $\chi_{2408}(2085,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} + \frac{3}{7} a^{9} - \frac{2}{7} a^{7} - \frac{2}{7} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{3} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{11} - \frac{2}{7} a^{9} - \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2}$, $\frac{1}{49} a^{12} - \frac{3}{49} a^{11} + \frac{3}{49} a^{10} - \frac{23}{49} a^{9} - \frac{18}{49} a^{8} + \frac{23}{49} a^{6} - \frac{3}{49} a^{5} + \frac{19}{49} a^{4} - \frac{6}{49} a^{3} + \frac{2}{7} a$, $\frac{1}{225004472072607099366125166801818734609} a^{13} - \frac{1603677392583638593136792771575566296}{225004472072607099366125166801818734609} a^{12} - \frac{1171749967697582404466766934248067628}{225004472072607099366125166801818734609} a^{11} + \frac{9647379472353947335487673031239539679}{225004472072607099366125166801818734609} a^{10} + \frac{1452065719518081629874759054701387411}{32143496010372442766589309543116962087} a^{9} - \frac{34200378703036208360875336243316028537}{225004472072607099366125166801818734609} a^{8} - \frac{102072472330932465879075770994266517969}{225004472072607099366125166801818734609} a^{7} - \frac{329043556408313239571966621978830000}{4591928001481777538084187077588137441} a^{6} + \frac{83831049772188805331672918989356212435}{225004472072607099366125166801818734609} a^{5} - \frac{87324120309357733108629763542535561605}{225004472072607099366125166801818734609} a^{4} - \frac{96509611755308282772372483119684645976}{225004472072607099366125166801818734609} a^{3} + \frac{2067830093242899262884753422850663330}{32143496010372442766589309543116962087} a^{2} + \frac{14223303037985682175204397342295751031}{32143496010372442766589309543116962087} a - \frac{1158680381347401370427663180987401366}{4591928001481777538084187077588137441}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{8516}$, which has order $545024$ (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:  \( 35991.64185055774 \) (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

\(\Q(\sqrt{-14}) \), 7.7.6321363049.1

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 R ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ R ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$2$2.14.21.34$x^{14} + 4 x^{13} + 8 x^{12} + 4 x^{11} + 5 x^{10} + 8 x^{9} - 6 x^{8} - 6 x^{7} + x^{6} + 6 x^{5} + 2 x^{3} + 7 x^{2} + 6 x - 7$$2$$7$$21$$C_{14}$$[3]^{7}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
$43$43.14.12.1$x^{14} + 3569 x^{7} + 4043763$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$