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Properties

Label 18.0.22259864297...1875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 5^{9}\cdot 7^{9}$
Root discriminant $25.60$
Ramified primes $3, 5, 7$
Class number $6$
Class group $[6]$
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![1299, 360, 1176, 4937, 501, 3615, 4972, -1593, 2838, -2224, 2337, -1224, 730, -414, 216, -76, 24, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 24*x^16 - 76*x^15 + 216*x^14 - 414*x^13 + 730*x^12 - 1224*x^11 + 2337*x^10 - 2224*x^9 + 2838*x^8 - 1593*x^7 + 4972*x^6 + 3615*x^5 + 501*x^4 + 4937*x^3 + 1176*x^2 + 360*x + 1299)
 
gp: K = bnfinit(x^18 - 6*x^17 + 24*x^16 - 76*x^15 + 216*x^14 - 414*x^13 + 730*x^12 - 1224*x^11 + 2337*x^10 - 2224*x^9 + 2838*x^8 - 1593*x^7 + 4972*x^6 + 3615*x^5 + 501*x^4 + 4937*x^3 + 1176*x^2 + 360*x + 1299, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 24 x^{16} - 76 x^{15} + 216 x^{14} - 414 x^{13} + 730 x^{12} - 1224 x^{11} + 2337 x^{10} - 2224 x^{9} + 2838 x^{8} - 1593 x^{7} + 4972 x^{6} + 3615 x^{5} + 501 x^{4} + 4937 x^{3} + 1176 x^{2} + 360 x + 1299 \)

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:  \(-22259864297551634701171875=-\,3^{24}\cdot 5^{9}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.60$
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:  $3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{7}$, $\frac{1}{18} a^{14} + \frac{1}{9} a^{13} + \frac{1}{18} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{7}{18} a^{8} - \frac{4}{9} a^{7} - \frac{2}{9} a^{6} + \frac{2}{9} a^{5} + \frac{5}{18} a^{4} - \frac{5}{18} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{54} a^{15} - \frac{1}{18} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{10} + \frac{7}{54} a^{9} - \frac{2}{9} a^{8} + \frac{2}{9} a^{7} - \frac{2}{9} a^{6} + \frac{1}{6} a^{5} + \frac{7}{18} a^{4} + \frac{19}{54} a^{3} - \frac{4}{9} a^{2} - \frac{1}{6} a + \frac{4}{9}$, $\frac{1}{42876} a^{16} + \frac{235}{42876} a^{15} - \frac{74}{3573} a^{14} + \frac{53}{14292} a^{13} + \frac{1165}{14292} a^{12} + \frac{3}{397} a^{11} + \frac{2377}{42876} a^{10} + \frac{5785}{42876} a^{9} - \frac{4703}{14292} a^{8} + \frac{92}{3573} a^{7} - \frac{373}{1588} a^{6} + \frac{578}{1191} a^{5} - \frac{19667}{42876} a^{4} - \frac{4205}{10719} a^{3} - \frac{3313}{7146} a^{2} + \frac{6725}{14292} a + \frac{5159}{14292}$, $\frac{1}{403443949151648034906246108} a^{17} - \frac{764248655117285153372}{100860987287912008726561527} a^{16} - \frac{180354822704335243756547}{403443949151648034906246108} a^{15} - \frac{3452160453724601471471789}{134481316383882678302082036} a^{14} - \frac{4534855497742417401757535}{33620329095970669575520509} a^{13} - \frac{13114500617010962911826993}{134481316383882678302082036} a^{12} + \frac{3085595981782147855214761}{403443949151648034906246108} a^{11} - \frac{2206228232008628082338791}{201721974575824017453123054} a^{10} - \frac{19848626249397945412937377}{201721974575824017453123054} a^{9} - \frac{39153235127049741526309561}{134481316383882678302082036} a^{8} - \frac{6426397826506655544290969}{14942368487098075366898004} a^{7} + \frac{3597631040949609649927771}{14942368487098075366898004} a^{6} + \frac{79573880303911858542822391}{403443949151648034906246108} a^{5} + \frac{124017458421793463206026577}{403443949151648034906246108} a^{4} - \frac{63602564312118288800174779}{201721974575824017453123054} a^{3} + \frac{5420386540154928917597761}{14942368487098075366898004} a^{2} + \frac{19259142889355617319208511}{67240658191941339151041018} a - \frac{62385226480871543123313619}{134481316383882678302082036}$

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

Class group and class number

$C_{6}$, which has order $6$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 142170.752754 \)
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

\(\Q(\sqrt{-35}) \), 3.1.2835.1 x3, \(\Q(\zeta_{9})^+\), 6.0.281302875.1, 6.0.3472875.1 x2, 6.0.281302875.3, 9.3.22785532875.2 x3

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

Sibling fields

Degree 6 sibling: 6.0.3472875.1
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$