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Properties

Label 18.0.17909931017...5107.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,37^{12}\cdot 67^{9}$
Root discriminant $90.89$
Ramified primes $37, 67$
Class number $44032$ (GRH)
Class group $[2, 2, 2, 2, 8, 344]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 43, -295, 1771, -10244, 33797, -39753, 59244, -35038, 48833, -23677, 24354, -4613, 1392, -170, 60, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 60*x^16 - 170*x^15 + 1392*x^14 - 4613*x^13 + 24354*x^12 - 23677*x^11 + 48833*x^10 - 35038*x^9 + 59244*x^8 - 39753*x^7 + 33797*x^6 - 10244*x^5 + 1771*x^4 - 295*x^3 + 43*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^18 - 7*x^17 + 60*x^16 - 170*x^15 + 1392*x^14 - 4613*x^13 + 24354*x^12 - 23677*x^11 + 48833*x^10 - 35038*x^9 + 59244*x^8 - 39753*x^7 + 33797*x^6 - 10244*x^5 + 1771*x^4 - 295*x^3 + 43*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 60 x^{16} - 170 x^{15} + 1392 x^{14} - 4613 x^{13} + 24354 x^{12} - 23677 x^{11} + 48833 x^{10} - 35038 x^{9} + 59244 x^{8} - 39753 x^{7} + 33797 x^{6} - 10244 x^{5} + 1771 x^{4} - 295 x^{3} + 43 x^{2} - 8 x + 1 \)

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:  \(-179099310176045734691646526162025107=-\,37^{12}\cdot 67^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.89$
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:  $37, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1363} a^{15} - \frac{135}{1363} a^{14} - \frac{499}{1363} a^{13} - \frac{182}{1363} a^{12} + \frac{86}{1363} a^{11} - \frac{477}{1363} a^{10} - \frac{509}{1363} a^{9} + \frac{594}{1363} a^{8} + \frac{436}{1363} a^{7} - \frac{414}{1363} a^{6} + \frac{111}{1363} a^{5} + \frac{518}{1363} a^{4} + \frac{316}{1363} a^{3} + \frac{439}{1363} a^{2} - \frac{264}{1363} a + \frac{176}{1363}$, $\frac{1}{42253} a^{16} - \frac{12}{42253} a^{15} + \frac{14245}{42253} a^{14} - \frac{17943}{42253} a^{13} - \frac{5944}{42253} a^{12} + \frac{12827}{42253} a^{11} - \frac{14201}{42253} a^{10} - \frac{3404}{42253} a^{9} + \frac{12163}{42253} a^{8} + \frac{57}{42253} a^{7} - \frac{364}{1363} a^{6} + \frac{1904}{42253} a^{5} - \frac{5483}{42253} a^{4} - \frac{16576}{42253} a^{3} + \frac{14206}{42253} a^{2} + \frac{585}{1363} a + \frac{12107}{42253}$, $\frac{1}{29707534199807400418280526569} a^{17} + \frac{193602208962201737745741}{29707534199807400418280526569} a^{16} + \frac{9954924776602142883909698}{29707534199807400418280526569} a^{15} + \frac{3825495566074968431659986063}{29707534199807400418280526569} a^{14} + \frac{10333069659350867155456918496}{29707534199807400418280526569} a^{13} + \frac{11012808812179769161389095175}{29707534199807400418280526569} a^{12} - \frac{9915899388090628718073233483}{29707534199807400418280526569} a^{11} + \frac{10255124378013430468823596587}{29707534199807400418280526569} a^{10} - \frac{8243733830691599103835774014}{29707534199807400418280526569} a^{9} + \frac{9113404155571082404965123220}{29707534199807400418280526569} a^{8} - \frac{10079007309389786868799487619}{29707534199807400418280526569} a^{7} - \frac{348531502670338907374437741}{29707534199807400418280526569} a^{6} + \frac{472813969516442407821387208}{29707534199807400418280526569} a^{5} + \frac{8602147778106207010660742430}{29707534199807400418280526569} a^{4} + \frac{897794653375436072621173362}{29707534199807400418280526569} a^{3} - \frac{12555736724441240379356603323}{29707534199807400418280526569} a^{2} + \frac{3347968422830391340545599194}{29707534199807400418280526569} a - \frac{13467964112970475537831603186}{29707534199807400418280526569}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{8}\times C_{344}$, which has order $44032$ (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:  \( -1 \) (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:  \( 424333.045004 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-67}) \), 3.1.91723.1 x3, 3.3.1369.1, 6.0.563678284843.1, 6.0.411744547.2 x2, 6.0.563678284843.4, 9.3.771675571950067.1 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 6 sibling: 6.0.411744547.2
Degree 9 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$37$37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$67$67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$