↑

Properties

Label 14.0.11027596817...6368.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 197^{13}$
Root discriminant $270.15$
Ramified primes $2, 197$
Class number $43760$ (GRH)
Class group $[2, 2, 2, 5470]$ (GRH)
Galois group $C_{14}$ (as 14T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18567053, 0, 45764873, 0, 26038278, 0, 5284722, 0, 415079, 0, 13790, 0, 197, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 197*x^12 + 13790*x^10 + 415079*x^8 + 5284722*x^6 + 26038278*x^4 + 45764873*x^2 + 18567053)
 
gp: K = bnfinit(x^14 + 197*x^12 + 13790*x^10 + 415079*x^8 + 5284722*x^6 + 26038278*x^4 + 45764873*x^2 + 18567053, 1)
 

Normalized defining polynomial

\( x^{14} + 197 x^{12} + 13790 x^{10} + 415079 x^{8} + 5284722 x^{6} + 26038278 x^{4} + 45764873 x^{2} + 18567053 \)

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:  \(-11027596817823724526645960905146368=-\,2^{14}\cdot 197^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $270.15$
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:  $2, 197$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(788=2^{2}\cdot 197\)
Dirichlet character group:    $\lbrace$$\chi_{788}(1,·)$, $\chi_{788}(83,·)$, $\chi_{788}(555,·)$, $\chi_{788}(487,·)$, $\chi_{788}(233,·)$, $\chi_{788}(427,·)$, $\chi_{788}(301,·)$, $\chi_{788}(19,·)$, $\chi_{788}(203,·)$, $\chi_{788}(585,·)$, $\chi_{788}(787,·)$, $\chi_{788}(705,·)$, $\chi_{788}(769,·)$, $\chi_{788}(361,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19} a^{10} + \frac{5}{19} a^{8} + \frac{4}{19} a^{6} - \frac{8}{19} a^{4} - \frac{2}{19} a^{2} + \frac{6}{19}$, $\frac{1}{19} a^{11} + \frac{5}{19} a^{9} + \frac{4}{19} a^{7} - \frac{8}{19} a^{5} - \frac{2}{19} a^{3} + \frac{6}{19} a$, $\frac{1}{818969980493651673131} a^{12} - \frac{6756830299676330042}{818969980493651673131} a^{10} - \frac{58091300276423903588}{818969980493651673131} a^{8} - \frac{308301926875320931055}{818969980493651673131} a^{6} - \frac{338018110015539474741}{818969980493651673131} a^{4} + \frac{332667865565215253545}{818969980493651673131} a^{2} + \frac{306449032699621177719}{818969980493651673131}$, $\frac{1}{251423784011551063651217} a^{13} - \frac{2765392554067766176378}{251423784011551063651217} a^{11} + \frac{47571478617907002349557}{251423784011551063651217} a^{9} + \frac{18140074475823779916317}{251423784011551063651217} a^{7} - \frac{105209279296386830039358}{251423784011551063651217} a^{5} + \frac{55807108123214147007208}{251423784011551063651217} a^{3} + \frac{22246223773292710736860}{251423784011551063651217} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{5470}$, which has order $43760$ (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:  \( 1553055.199048291 \) (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

\(\Q(\sqrt{-197}) \), 7.7.58451728309129.1

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 ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$2$2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
$197$197.14.13.1$x^{14} - 197$$14$$1$$13$$C_{14}$$[\ ]_{14}$