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Properties

Label 14.0.17670347510...2983.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{7}\cdot 337^{12}$
Root discriminant $388.23$
Ramified primes $7, 337$
Class number $149597$ (GRH)
Class group $[7, 21371]$ (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![1015807027, -1872983259, 1807246866, -995551231, 267969411, 6620102, -24009340, 4513593, 472092, -239968, 16962, 2173, -265, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 - 265*x^12 + 2173*x^11 + 16962*x^10 - 239968*x^9 + 472092*x^8 + 4513593*x^7 - 24009340*x^6 + 6620102*x^5 + 267969411*x^4 - 995551231*x^3 + 1807246866*x^2 - 1872983259*x + 1015807027)
 
gp: K = bnfinit(x^14 - 5*x^13 - 265*x^12 + 2173*x^11 + 16962*x^10 - 239968*x^9 + 472092*x^8 + 4513593*x^7 - 24009340*x^6 + 6620102*x^5 + 267969411*x^4 - 995551231*x^3 + 1807246866*x^2 - 1872983259*x + 1015807027, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} - 265 x^{12} + 2173 x^{11} + 16962 x^{10} - 239968 x^{9} + 472092 x^{8} + 4513593 x^{7} - 24009340 x^{6} + 6620102 x^{5} + 267969411 x^{4} - 995551231 x^{3} + 1807246866 x^{2} - 1872983259 x + 1015807027 \)

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:  \(-1767034751033661210807027893768732983=-\,7^{7}\cdot 337^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $388.23$
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:  $7, 337$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2359=7\cdot 337\)
Dirichlet character group:    $\lbrace$$\chi_{2359}(512,·)$, $\chi_{2359}(1,·)$, $\chi_{2359}(295,·)$, $\chi_{2359}(8,·)$, $\chi_{2359}(1737,·)$, $\chi_{2359}(1063,·)$, $\chi_{2359}(64,·)$, $\chi_{2359}(1427,·)$, $\chi_{2359}(2101,·)$, $\chi_{2359}(1749,·)$, $\chi_{2359}(1686,·)$, $\chi_{2359}(1980,·)$, $\chi_{2359}(1693,·)$, $\chi_{2359}(2197,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{328261781} a^{12} - \frac{47270060}{328261781} a^{11} + \frac{125457172}{328261781} a^{10} + \frac{124763220}{328261781} a^{9} + \frac{54463744}{328261781} a^{8} + \frac{87916992}{328261781} a^{7} - \frac{128018647}{328261781} a^{6} + \frac{26580228}{328261781} a^{5} + \frac{83014523}{328261781} a^{4} - \frac{100819341}{328261781} a^{3} + \frac{74266140}{328261781} a^{2} + \frac{161597073}{328261781} a + \frac{145133361}{328261781}$, $\frac{1}{584491049559809924206514618994116225379837559} a^{13} - \frac{519550099647716992010855664751112243}{584491049559809924206514618994116225379837559} a^{12} + \frac{150742095121769570739933210892204433248486163}{584491049559809924206514618994116225379837559} a^{11} - \frac{13385092329343758599339508105710068948523306}{584491049559809924206514618994116225379837559} a^{10} + \frac{131381800131383790729058039278577373354439872}{584491049559809924206514618994116225379837559} a^{9} - \frac{2539343477250421704137494324293899818210571}{9906627958640846172991773203290105514912501} a^{8} - \frac{50204776302343727167113288334672123851433232}{584491049559809924206514618994116225379837559} a^{7} - \frac{235535148837464450655981946459770665313515063}{584491049559809924206514618994116225379837559} a^{6} - \frac{76868521219757028665013400241289889107376694}{584491049559809924206514618994116225379837559} a^{5} + \frac{37681459930735033057207366525965095575972212}{584491049559809924206514618994116225379837559} a^{4} + \frac{246814894624822153021376929069512465262290994}{584491049559809924206514618994116225379837559} a^{3} + \frac{44709084191292372516600245967647936995011294}{584491049559809924206514618994116225379837559} a^{2} + \frac{237825346637907693318023268729709025847093281}{584491049559809924206514618994116225379837559} a + \frac{69960238765952584524846397405230318596414696}{584491049559809924206514618994116225379837559}$

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

Class group and class number

$C_{7}\times C_{21371}$, which has order $149597$ (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:  \( 5486180.267403393 \) (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

\(\Q(\sqrt{-7}) \), 7.7.1464803622199009.1

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 ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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}$
337Data not computed