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Properties

Label 18.18.7207333436...6957.1
Degree $18$
Signature $[18, 0]$
Discriminant $7^{15}\cdot 19^{15}$
Root discriminant $58.87$
Ramified primes $7, 19$
Class number $1$ (GRH)
Class group Trivial (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![-189, 2835, -14301, 24745, 14070, -79772, 32656, 77678, -51037, -34628, 27250, 7833, -6901, -903, 860, 50, -49, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 49*x^16 + 50*x^15 + 860*x^14 - 903*x^13 - 6901*x^12 + 7833*x^11 + 27250*x^10 - 34628*x^9 - 51037*x^8 + 77678*x^7 + 32656*x^6 - 79772*x^5 + 14070*x^4 + 24745*x^3 - 14301*x^2 + 2835*x - 189)
 
gp: K = bnfinit(x^18 - x^17 - 49*x^16 + 50*x^15 + 860*x^14 - 903*x^13 - 6901*x^12 + 7833*x^11 + 27250*x^10 - 34628*x^9 - 51037*x^8 + 77678*x^7 + 32656*x^6 - 79772*x^5 + 14070*x^4 + 24745*x^3 - 14301*x^2 + 2835*x - 189, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 49 x^{16} + 50 x^{15} + 860 x^{14} - 903 x^{13} - 6901 x^{12} + 7833 x^{11} + 27250 x^{10} - 34628 x^{9} - 51037 x^{8} + 77678 x^{7} + 32656 x^{6} - 79772 x^{5} + 14070 x^{4} + 24745 x^{3} - 14301 x^{2} + 2835 x - 189 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(72073334364588888282643007986957=7^{15}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.87$
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$$\chi_{133}(64,·)$, $\chi_{133}(1,·)$, $\chi_{133}(75,·)$, $\chi_{133}(132,·)$, $\chi_{133}(69,·)$, $\chi_{133}(11,·)$, $\chi_{133}(12,·)$, $\chi_{133}(27,·)$, $\chi_{133}(58,·)$, $\chi_{133}(30,·)$, $\chi_{133}(31,·)$, $\chi_{133}(102,·)$, $\chi_{133}(39,·)$, $\chi_{133}(106,·)$, $\chi_{133}(103,·)$, $\chi_{133}(94,·)$, $\chi_{133}(121,·)$, $\chi_{133}(122,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{797} a^{15} - \frac{252}{797} a^{14} + \frac{79}{797} a^{13} + \frac{267}{797} a^{12} + \frac{49}{797} a^{11} - \frac{58}{797} a^{10} + \frac{295}{797} a^{9} - \frac{247}{797} a^{8} + \frac{98}{797} a^{7} + \frac{105}{797} a^{6} + \frac{284}{797} a^{5} - \frac{70}{797} a^{4} + \frac{375}{797} a^{3} - \frac{165}{797} a^{2} - \frac{279}{797} a + \frac{371}{797}$, $\frac{1}{2101689} a^{16} - \frac{742}{2101689} a^{15} + \frac{871145}{2101689} a^{14} - \frac{40834}{2101689} a^{13} + \frac{602459}{2101689} a^{12} - \frac{70065}{233521} a^{11} - \frac{227122}{2101689} a^{10} + \frac{67565}{700563} a^{9} + \frac{395296}{2101689} a^{8} + \frac{174448}{2101689} a^{7} - \frac{868888}{2101689} a^{6} - \frac{774439}{2101689} a^{5} - \frac{991064}{2101689} a^{4} - \frac{827891}{2101689} a^{3} + \frac{40301}{233521} a^{2} + \frac{61366}{2101689} a + \frac{276800}{700563}$, $\frac{1}{1492445198900019729159} a^{17} + \frac{216813724319387}{1492445198900019729159} a^{16} + \frac{726145657278168170}{1492445198900019729159} a^{15} - \frac{253020351258456691210}{1492445198900019729159} a^{14} + \frac{504995610217600982657}{1492445198900019729159} a^{13} - \frac{197304508836533255915}{497481732966673243053} a^{12} - \frac{263165313266920737664}{1492445198900019729159} a^{11} - \frac{72990761203280209859}{165827244322224414351} a^{10} + \frac{134819257976635988599}{1492445198900019729159} a^{9} - \frac{177482834432331978242}{1492445198900019729159} a^{8} - \frac{1263057185453183005}{1492445198900019729159} a^{7} + \frac{423478686368630609357}{1492445198900019729159} a^{6} - \frac{284080585913995412834}{1492445198900019729159} a^{5} - \frac{161367927063320474480}{1492445198900019729159} a^{4} + \frac{139015995669249026192}{497481732966673243053} a^{3} + \frac{541397194760582349070}{1492445198900019729159} a^{2} + \frac{2477354121541934998}{497481732966673243053} a - \frac{20735911804747184714}{165827244322224414351}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:  \( 15523352898.4 \) (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

\(\Q(\sqrt{133}) \), 3.3.361.1, 3.3.17689.1, 3.3.17689.2, \(\Q(\zeta_{7})^+\), 6.6.849301957.1, 6.6.41615795893.2, 6.6.41615795893.1, 6.6.115279213.1, 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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