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Properties

Label 14.0.87868184310...7024.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 3^{7}\cdot 7^{24}$
Root discriminant $137.67$
Ramified primes $2, 3, 7$
Class number $49798$ (GRH)
Class group $[49798]$ (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![17724049, 7397628, 5724964, 1471666, 949228, 216482, 98329, 10768, 6405, 854, 371, -42, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 42*x^11 + 371*x^10 + 854*x^9 + 6405*x^8 + 10768*x^7 + 98329*x^6 + 216482*x^5 + 949228*x^4 + 1471666*x^3 + 5724964*x^2 + 7397628*x + 17724049)
 
gp: K = bnfinit(x^14 - 42*x^11 + 371*x^10 + 854*x^9 + 6405*x^8 + 10768*x^7 + 98329*x^6 + 216482*x^5 + 949228*x^4 + 1471666*x^3 + 5724964*x^2 + 7397628*x + 17724049, 1)
 

Normalized defining polynomial

\( x^{14} - 42 x^{11} + 371 x^{10} + 854 x^{9} + 6405 x^{8} + 10768 x^{7} + 98329 x^{6} + 216482 x^{5} + 949228 x^{4} + 1471666 x^{3} + 5724964 x^{2} + 7397628 x + 17724049 \)

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:  \(-878681843101699928584328577024=-\,2^{21}\cdot 3^{7}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $137.67$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1176=2^{3}\cdot 3\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{1176}(1,·)$, $\chi_{1176}(197,·)$, $\chi_{1176}(673,·)$, $\chi_{1176}(841,·)$, $\chi_{1176}(29,·)$, $\chi_{1176}(1037,·)$, $\chi_{1176}(337,·)$, $\chi_{1176}(365,·)$, $\chi_{1176}(1009,·)$, $\chi_{1176}(533,·)$, $\chi_{1176}(169,·)$, $\chi_{1176}(505,·)$, $\chi_{1176}(701,·)$, $\chi_{1176}(869,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{589} a^{12} - \frac{111}{589} a^{11} - \frac{42}{589} a^{10} - \frac{169}{589} a^{9} + \frac{30}{589} a^{8} + \frac{44}{589} a^{7} - \frac{66}{589} a^{6} + \frac{99}{589} a^{5} - \frac{12}{31} a^{4} - \frac{284}{589} a^{3} - \frac{123}{589} a^{2} - \frac{4}{31} a - \frac{75}{589}$, $\frac{1}{216764782825250159263182599049673117} a^{13} + \frac{158758528468851758029443519743323}{216764782825250159263182599049673117} a^{12} - \frac{100510880590552091182494087311696464}{216764782825250159263182599049673117} a^{11} - \frac{105530484612907145974735893696417357}{216764782825250159263182599049673117} a^{10} - \frac{67599256724207599618995067456798235}{216764782825250159263182599049673117} a^{9} - \frac{24013546279439920761400504701920531}{216764782825250159263182599049673117} a^{8} - \frac{144052561052562755844966038564131}{216764782825250159263182599049673117} a^{7} - \frac{98080487117483788412995779158313826}{216764782825250159263182599049673117} a^{6} - \frac{69503080997565348742624824514951065}{216764782825250159263182599049673117} a^{5} + \frac{38793065925268242754591169105319118}{216764782825250159263182599049673117} a^{4} - \frac{4375883752840426698034258579574021}{11408672780276324171746452581561743} a^{3} + \frac{3013457838674931934076435848620969}{216764782825250159263182599049673117} a^{2} + \frac{31375685705060150437248714898774350}{216764782825250159263182599049673117} a - \frac{68154777104178861546047943877229421}{216764782825250159263182599049673117}$

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

Class group and class number

$C_{49798}$, which has order $49798$ (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

\(\Q(\sqrt{-6}) \), 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 R R ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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$2.14.21.33$x^{14} + 4 x^{13} + 4 x^{12} + 4 x^{11} - 3 x^{10} + 4 x^{9} - 2 x^{7} - x^{6} - 2 x^{5} + 2 x^{4} - 2 x^{3} + 3 x^{2} + 2 x + 1$$2$$7$$21$$C_{14}$$[3]^{7}$
$3$3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$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]$