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Properties

Label 18.0.10507848719...8823.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{15}\cdot 19^{12}$
Root discriminant $36.04$
Ramified primes $7, 19$
Class number $52$ (GRH)
Class group $[2, 26]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![117649, -100842, 103243, -86093, 74137, -61096, 50660, -12454, 9184, -2414, 1602, -406, 232, -28, 41, -6, 7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 7*x^16 - 6*x^15 + 41*x^14 - 28*x^13 + 232*x^12 - 406*x^11 + 1602*x^10 - 2414*x^9 + 9184*x^8 - 12454*x^7 + 50660*x^6 - 61096*x^5 + 74137*x^4 - 86093*x^3 + 103243*x^2 - 100842*x + 117649)
 
gp: K = bnfinit(x^18 - x^17 + 7*x^16 - 6*x^15 + 41*x^14 - 28*x^13 + 232*x^12 - 406*x^11 + 1602*x^10 - 2414*x^9 + 9184*x^8 - 12454*x^7 + 50660*x^6 - 61096*x^5 + 74137*x^4 - 86093*x^3 + 103243*x^2 - 100842*x + 117649, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 7 x^{16} - 6 x^{15} + 41 x^{14} - 28 x^{13} + 232 x^{12} - 406 x^{11} + 1602 x^{10} - 2414 x^{9} + 9184 x^{8} - 12454 x^{7} + 50660 x^{6} - 61096 x^{5} + 74137 x^{4} - 86093 x^{3} + 103243 x^{2} - 100842 x + 117649 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-10507848719141112156676338823=-\,7^{15}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.04$
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:  \(133=7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{133}(64,·)$, $\chi_{133}(1,·)$, $\chi_{133}(68,·)$, $\chi_{133}(11,·)$, $\chi_{133}(83,·)$, $\chi_{133}(20,·)$, $\chi_{133}(87,·)$, $\chi_{133}(26,·)$, $\chi_{133}(30,·)$, $\chi_{133}(96,·)$, $\chi_{133}(102,·)$, $\chi_{133}(39,·)$, $\chi_{133}(106,·)$, $\chi_{133}(45,·)$, $\chi_{133}(115,·)$, $\chi_{133}(121,·)$, $\chi_{133}(58,·)$, $\chi_{133}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{4} a^{7} + \frac{1}{4}$, $\frac{1}{4} a^{8} + \frac{1}{4} a$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{5}$, $\frac{1}{4393564} a^{13} + \frac{515563}{4393564} a^{12} - \frac{11427}{627652} a^{11} - \frac{121799}{4393564} a^{10} - \frac{44367}{4393564} a^{9} - \frac{21137}{627652} a^{8} + \frac{127555}{4393564} a^{7} + \frac{140643}{627652} a^{6} + \frac{159095}{4393564} a^{5} - \frac{2147613}{4393564} a^{4} - \frac{298373}{627652} a^{3} + \frac{202117}{4393564} a^{2} - \frac{307299}{4393564} a + \frac{168401}{627652}$, $\frac{1}{123019792} a^{14} - \frac{1}{15377474} a^{13} + \frac{108879}{1098391} a^{12} + \frac{118451}{7688737} a^{11} + \frac{610095}{7688737} a^{10} - \frac{189069}{4393564} a^{9} - \frac{945955}{7688737} a^{8} + \frac{798247}{8787128} a^{7} + \frac{6547323}{15377474} a^{6} - \frac{1727134}{7688737} a^{5} - \frac{306130}{1098391} a^{4} + \frac{3162631}{7688737} a^{3} - \frac{12486535}{30754948} a^{2} + \frac{27153}{156913} a + \frac{106145}{2510608}$, $\frac{1}{861138544} a^{15} - \frac{1}{861138544} a^{14} + \frac{1}{15377474} a^{13} - \frac{3609028}{53821159} a^{12} - \frac{4728845}{215284636} a^{11} + \frac{3679227}{30754948} a^{10} + \frac{3141517}{107642318} a^{9} - \frac{7342189}{61509896} a^{8} - \frac{50855613}{430569272} a^{7} + \frac{32311413}{107642318} a^{6} + \frac{942569}{7688737} a^{5} + \frac{10260059}{215284636} a^{4} + \frac{62239725}{215284636} a^{3} - \frac{6074403}{15377474} a^{2} + \frac{1920025}{17574256} a + \frac{739025}{2510608}$, $\frac{1}{6027969808} a^{16} - \frac{1}{6027969808} a^{15} + \frac{1}{861138544} a^{14} + \frac{85}{753496226} a^{13} - \frac{104572715}{1506992452} a^{12} - \frac{7748371}{215284636} a^{11} + \frac{180299723}{1506992452} a^{10} + \frac{5934165}{430569272} a^{9} - \frac{144275289}{3013984904} a^{8} - \frac{96278563}{3013984904} a^{7} - \frac{29156551}{107642318} a^{6} - \frac{551805907}{1506992452} a^{5} - \frac{467244573}{1506992452} a^{4} - \frac{60922755}{215284636} a^{3} + \frac{55582885}{123019792} a^{2} + \frac{3161609}{17574256} a - \frac{1129485}{2510608}$, $\frac{1}{42195788656} a^{17} - \frac{1}{42195788656} a^{16} + \frac{1}{6027969808} a^{15} - \frac{3}{21097894328} a^{14} - \frac{295}{5274473582} a^{13} - \frac{44140413}{376748113} a^{12} - \frac{1223331051}{10548947164} a^{11} + \frac{204991177}{3013984904} a^{10} - \frac{1771111255}{21097894328} a^{9} - \frac{1471420443}{21097894328} a^{8} + \frac{63522899}{1506992452} a^{7} - \frac{2596107417}{5274473582} a^{6} - \frac{378796001}{2637236791} a^{5} - \frac{490058973}{1506992452} a^{4} + \frac{180604733}{861138544} a^{3} + \frac{41051221}{123019792} a^{2} - \frac{255227}{2510608} a - \frac{537357}{1255304}$

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

Class group and class number

$C_{2}\times C_{26}$, which has order $52$ (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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{226365}{21097894328} a^{17} + \frac{75455}{42195788656} a^{16} + \frac{377275}{6027969808} a^{15} + \frac{980915}{42195788656} a^{14} + \frac{3848205}{10548947164} a^{13} + \frac{290071}{1506992452} a^{12} + \frac{22561045}{10548947164} a^{11} - \frac{2188195}{1506992452} a^{10} + \frac{255415175}{21097894328} a^{9} - \frac{123368925}{21097894328} a^{8} + \frac{205916695}{3013984904} a^{7} - \frac{196862095}{10548947164} a^{6} + \frac{4141145837}{10548947164} a^{5} - \frac{4602755}{215284636} a^{4} + \frac{1886375}{61509896} a^{3} + \frac{75455}{17574256} a^{2} + \frac{75455}{2510608} a + \frac{528185}{2510608} \) (order $14$)
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:  \( 833965.243856 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.3.17689.1, 3.3.17689.2, \(\Q(\zeta_{7})^+\), 3.3.361.1, 6.0.2190305047.1, 6.0.2190305047.2, \(\Q(\zeta_{7})\), 6.0.44700103.1, 9.9.5534900853769.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
19Data not computed