↑

Properties

Label 14.0.65230065177...2247.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{24}\cdot 23^{7}$
Root discriminant $134.77$
Ramified primes $7, 23$
Class number $19659$ (GRH)
Class group $[19659]$ (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![19486321, -12694353, 8318800, -3648323, 1703975, -555058, 173194, -44565, 12670, -2058, 154, 7, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 21*x^12 + 7*x^11 + 154*x^10 - 2058*x^9 + 12670*x^8 - 44565*x^7 + 173194*x^6 - 555058*x^5 + 1703975*x^4 - 3648323*x^3 + 8318800*x^2 - 12694353*x + 19486321)
 
gp: K = bnfinit(x^14 - 7*x^13 + 21*x^12 + 7*x^11 + 154*x^10 - 2058*x^9 + 12670*x^8 - 44565*x^7 + 173194*x^6 - 555058*x^5 + 1703975*x^4 - 3648323*x^3 + 8318800*x^2 - 12694353*x + 19486321, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 21 x^{12} + 7 x^{11} + 154 x^{10} - 2058 x^{9} + 12670 x^{8} - 44565 x^{7} + 173194 x^{6} - 555058 x^{5} + 1703975 x^{4} - 3648323 x^{3} + 8318800 x^{2} - 12694353 x + 19486321 \)

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:  \(-652300651772147469026072062247=-\,7^{24}\cdot 23^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $134.77$
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:  $7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1127=7^{2}\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{1127}(1,·)$, $\chi_{1127}(162,·)$, $\chi_{1127}(323,·)$, $\chi_{1127}(484,·)$, $\chi_{1127}(645,·)$, $\chi_{1127}(806,·)$, $\chi_{1127}(967,·)$, $\chi_{1127}(22,·)$, $\chi_{1127}(183,·)$, $\chi_{1127}(344,·)$, $\chi_{1127}(505,·)$, $\chi_{1127}(666,·)$, $\chi_{1127}(827,·)$, $\chi_{1127}(988,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{31} a^{10} + \frac{9}{31} a^{9} - \frac{6}{31} a^{8} + \frac{3}{31} a^{7} + \frac{11}{31} a^{6} + \frac{3}{31} a^{5} + \frac{2}{31} a^{4} + \frac{10}{31} a^{3} + \frac{11}{31} a^{2} + \frac{10}{31} a$, $\frac{1}{31} a^{11} + \frac{6}{31} a^{9} - \frac{5}{31} a^{8} + \frac{15}{31} a^{7} - \frac{3}{31} a^{6} + \frac{6}{31} a^{5} - \frac{8}{31} a^{4} + \frac{14}{31} a^{3} + \frac{4}{31} a^{2} + \frac{3}{31} a$, $\frac{1}{589} a^{12} - \frac{9}{589} a^{11} - \frac{1}{589} a^{10} - \frac{277}{589} a^{9} + \frac{102}{589} a^{8} - \frac{283}{589} a^{7} - \frac{230}{589} a^{6} - \frac{238}{589} a^{5} - \frac{176}{589} a^{4} - \frac{15}{31} a^{3} + \frac{293}{589} a^{2} - \frac{35}{589} a + \frac{7}{19}$, $\frac{1}{119498557037247221462377659843307} a^{13} - \frac{21171491709700295636341060489}{119498557037247221462377659843307} a^{12} - \frac{510617294722728787847180411071}{119498557037247221462377659843307} a^{11} - \frac{240041903321169532876287762303}{119498557037247221462377659843307} a^{10} - \frac{21836611061260366731519677430115}{119498557037247221462377659843307} a^{9} - \frac{50115114132900643540147757306327}{119498557037247221462377659843307} a^{8} + \frac{57397883941821085806232431443751}{119498557037247221462377659843307} a^{7} + \frac{4376815724103568861851630845170}{119498557037247221462377659843307} a^{6} - \frac{7543948309392285930185995218828}{119498557037247221462377659843307} a^{5} - \frac{47424367847492490135950691652961}{119498557037247221462377659843307} a^{4} + \frac{55345190169601230503301807600484}{119498557037247221462377659843307} a^{3} + \frac{38521795653165302595113900470744}{119498557037247221462377659843307} a^{2} - \frac{27462738557882809113511524192893}{119498557037247221462377659843307} a - \frac{255466179587515296375686628069}{3854792162491845853625085801397}$

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

Class group and class number

$C_{19659}$, which has order $19659$ (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

\(\Q(\sqrt{-23}) \), 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$7$7.14.24.53$x^{14} + 931 x^{13} + 2310 x^{12} + 903 x^{11} + 392 x^{10} + 2198 x^{9} + 2296 x^{8} + 1485 x^{7} + 637 x^{6} + 1295 x^{5} + 2303 x^{4} + 1449 x^{3} + 1316 x^{2} + 2219 x + 2383$$7$$2$$24$$C_{14}$$[2]^{2}$
$23$23.14.7.2$x^{14} - 148035889 x^{2} + 27238603576$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$