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Properties

Label 14.0.66016163688...9375.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,5^{7}\cdot 7^{7}\cdot 29^{13}$
Root discriminant $134.89$
Ramified primes $5, 7, 29$
Class number $95104$ (GRH)
Class group $[2, 2, 2, 2, 5944]$ (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![476400689, -451469716, 444796489, -163620583, 74493972, -22664583, 11017281, -951747, 747078, -21196, 20946, -249, 248, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 248*x^12 - 249*x^11 + 20946*x^10 - 21196*x^9 + 747078*x^8 - 951747*x^7 + 11017281*x^6 - 22664583*x^5 + 74493972*x^4 - 163620583*x^3 + 444796489*x^2 - 451469716*x + 476400689)
 
gp: K = bnfinit(x^14 - x^13 + 248*x^12 - 249*x^11 + 20946*x^10 - 21196*x^9 + 747078*x^8 - 951747*x^7 + 11017281*x^6 - 22664583*x^5 + 74493972*x^4 - 163620583*x^3 + 444796489*x^2 - 451469716*x + 476400689, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 248 x^{12} - 249 x^{11} + 20946 x^{10} - 21196 x^{9} + 747078 x^{8} - 951747 x^{7} + 11017281 x^{6} - 22664583 x^{5} + 74493972 x^{4} - 163620583 x^{3} + 444796489 x^{2} - 451469716 x + 476400689 \)

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:  \(-660161636887192665823095859375=-\,5^{7}\cdot 7^{7}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $134.89$
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:  $5, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1015=5\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1015}(1,·)$, $\chi_{1015}(34,·)$, $\chi_{1015}(419,·)$, $\chi_{1015}(36,·)$, $\chi_{1015}(806,·)$, $\chi_{1015}(874,·)$, $\chi_{1015}(141,·)$, $\chi_{1015}(209,·)$, $\chi_{1015}(979,·)$, $\chi_{1015}(596,·)$, $\chi_{1015}(981,·)$, $\chi_{1015}(1014,·)$, $\chi_{1015}(281,·)$, $\chi_{1015}(734,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{41} a^{12} - \frac{15}{41} a^{11} - \frac{1}{41} a^{10} + \frac{8}{41} a^{9} + \frac{14}{41} a^{8} - \frac{13}{41} a^{7} + \frac{5}{41} a^{6} + \frac{20}{41} a^{5} + \frac{15}{41} a^{4} + \frac{11}{41} a^{3} + \frac{19}{41} a^{2} + \frac{18}{41} a$, $\frac{1}{25063016337792776789283119643313166955227880296832491671} a^{13} + \frac{241958923196887106453470834115400108650890429395840391}{25063016337792776789283119643313166955227880296832491671} a^{12} - \frac{4074735438796101410812375531124661355577894085240525259}{25063016337792776789283119643313166955227880296832491671} a^{11} - \frac{3169480317281134084405972761068782358576425910185925763}{25063016337792776789283119643313166955227880296832491671} a^{10} - \frac{6020893980505479418239057950031581340685400408039059104}{25063016337792776789283119643313166955227880296832491671} a^{9} + \frac{10302397929474273485709177808522657428117718362816772653}{25063016337792776789283119643313166955227880296832491671} a^{8} + \frac{3121311501450921298811666062324951113891147181763352502}{25063016337792776789283119643313166955227880296832491671} a^{7} + \frac{2530154719431635888384649768302692929112285285781969426}{25063016337792776789283119643313166955227880296832491671} a^{6} - \frac{988822868856110083481264793052293078729947665526624057}{25063016337792776789283119643313166955227880296832491671} a^{5} + \frac{12362223365454728417215871297815789870069148877065953613}{25063016337792776789283119643313166955227880296832491671} a^{4} + \frac{9547305186853378455399753722523317512505623959006023197}{25063016337792776789283119643313166955227880296832491671} a^{3} - \frac{10861690114173934294447229031376401548867430623796522578}{25063016337792776789283119643313166955227880296832491671} a^{2} - \frac{12151588989193563476141518460338076707591371725217751785}{25063016337792776789283119643313166955227880296832491671} a + \frac{77765809454547012261056175420640570082319655596993464}{611293081409579921689832186422272364761655616995914431}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{5944}$, which has order $95104$ (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:  \( 6020.985100147561 \) (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

\(\Q(\sqrt{-1015}) \), 7.7.594823321.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ R R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$5$5.14.7.2$x^{14} - 15625 x^{2} + 156250$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$7$7.14.7.2$x^{14} - 686 x^{8} + 117649 x^{2} - 3294172$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$