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Properties

Label 14.0.17124904845...3339.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{24}\cdot 19^{7}$
Root discriminant $122.49$
Ramified primes $7, 19$
Class number $12992$ (GRH)
Class group $[2, 2, 2, 2, 2, 406]$ (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![5746151, 1399160, 1882601, 126, 290430, -15162, 50239, -5165, 3269, -987, 448, -35, 14, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 14*x^12 - 35*x^11 + 448*x^10 - 987*x^9 + 3269*x^8 - 5165*x^7 + 50239*x^6 - 15162*x^5 + 290430*x^4 + 126*x^3 + 1882601*x^2 + 1399160*x + 5746151)
 
gp: K = bnfinit(x^14 - 7*x^13 + 14*x^12 - 35*x^11 + 448*x^10 - 987*x^9 + 3269*x^8 - 5165*x^7 + 50239*x^6 - 15162*x^5 + 290430*x^4 + 126*x^3 + 1882601*x^2 + 1399160*x + 5746151, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 14 x^{12} - 35 x^{11} + 448 x^{10} - 987 x^{9} + 3269 x^{8} - 5165 x^{7} + 50239 x^{6} - 15162 x^{5} + 290430 x^{4} + 126 x^{3} + 1882601 x^{2} + 1399160 x + 5746151 \)

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:  \(-171249048453908271645616513339=-\,7^{24}\cdot 19^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $122.49$
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:  $7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(931=7^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{931}(512,·)$, $\chi_{931}(1,·)$, $\chi_{931}(645,·)$, $\chi_{931}(134,·)$, $\chi_{931}(778,·)$, $\chi_{931}(267,·)$, $\chi_{931}(911,·)$, $\chi_{931}(400,·)$, $\chi_{931}(113,·)$, $\chi_{931}(533,·)$, $\chi_{931}(246,·)$, $\chi_{931}(666,·)$, $\chi_{931}(379,·)$, $\chi_{931}(799,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{589} a^{12} - \frac{117}{589} a^{11} + \frac{283}{589} a^{10} + \frac{102}{589} a^{9} + \frac{142}{589} a^{8} - \frac{219}{589} a^{7} - \frac{284}{589} a^{6} - \frac{276}{589} a^{5} - \frac{174}{589} a^{4} + \frac{275}{589} a^{3} + \frac{158}{589} a^{2} + \frac{12}{31} a + \frac{14}{31}$, $\frac{1}{7143651410261227570763450186434727} a^{13} + \frac{4307334855886101817694671133436}{7143651410261227570763450186434727} a^{12} + \frac{1718733965783239411604222713757575}{7143651410261227570763450186434727} a^{11} - \frac{2188658915205361932006837437652349}{7143651410261227570763450186434727} a^{10} + \frac{3422295047133238157474167765455372}{7143651410261227570763450186434727} a^{9} - \frac{285538137919108621354636523993247}{7143651410261227570763450186434727} a^{8} + \frac{594331784527807444858397990850880}{7143651410261227570763450186434727} a^{7} - \frac{1896735610943177146573129383252058}{7143651410261227570763450186434727} a^{6} + \frac{2231039655211338800763449318620147}{7143651410261227570763450186434727} a^{5} - \frac{761846555509957677334422305070390}{7143651410261227570763450186434727} a^{4} - \frac{3394178260341053206234068570775989}{7143651410261227570763450186434727} a^{3} - \frac{474481290281817088317478774968300}{7143651410261227570763450186434727} a^{2} - \frac{87406940049951799003185557748245}{375981653171643556355971062443933} a - \frac{131441830667989327441490276803379}{375981653171643556355971062443933}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{406}$, which has order $12992$ (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

\(\Q(\sqrt{-19}) \), 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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$