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Properties

Label 18.0.24370619933...6643.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 13^{12}$
Root discriminant $19.92$
Ramified primes $3, 13$
Class number $1$
Class group Trivial
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![25, -15, 264, -1187, 3858, -7140, 8797, -7785, 5337, -3127, 1956, -1473, 1114, -711, 369, -150, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 150*x^15 + 369*x^14 - 711*x^13 + 1114*x^12 - 1473*x^11 + 1956*x^10 - 3127*x^9 + 5337*x^8 - 7785*x^7 + 8797*x^6 - 7140*x^5 + 3858*x^4 - 1187*x^3 + 264*x^2 - 15*x + 25)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 150*x^15 + 369*x^14 - 711*x^13 + 1114*x^12 - 1473*x^11 + 1956*x^10 - 3127*x^9 + 5337*x^8 - 7785*x^7 + 8797*x^6 - 7140*x^5 + 3858*x^4 - 1187*x^3 + 264*x^2 - 15*x + 25, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 150 x^{15} + 369 x^{14} - 711 x^{13} + 1114 x^{12} - 1473 x^{11} + 1956 x^{10} - 3127 x^{9} + 5337 x^{8} - 7785 x^{7} + 8797 x^{6} - 7140 x^{5} + 3858 x^{4} - 1187 x^{3} + 264 x^{2} - 15 x + 25 \)

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:  \(-243706199334710775656643=-\,3^{21}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.92$
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:  $3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{15} a^{11} - \frac{1}{15} a^{10} + \frac{1}{15} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{15} a^{5} + \frac{7}{15} a^{4} + \frac{1}{3} a^{3} - \frac{2}{15} a^{2} - \frac{4}{15} a - \frac{1}{3}$, $\frac{1}{15} a^{12} - \frac{2}{15} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{15} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{3}$, $\frac{1}{15} a^{13} - \frac{2}{15} a^{10} - \frac{2}{15} a^{9} + \frac{1}{5} a^{8} - \frac{1}{15} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{15} a^{3} + \frac{2}{5} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{15} a^{14} + \frac{1}{15} a^{10} - \frac{7}{15} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{7}{15} a^{5} - \frac{7}{15} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{7}{15} a$, $\frac{1}{39195} a^{15} - \frac{27}{871} a^{14} - \frac{1094}{39195} a^{13} - \frac{17}{7839} a^{12} + \frac{1258}{39195} a^{11} + \frac{4291}{39195} a^{10} - \frac{979}{13065} a^{9} - \frac{4769}{13065} a^{8} - \frac{10378}{39195} a^{7} - \frac{14}{117} a^{6} + \frac{14789}{39195} a^{5} + \frac{257}{1005} a^{4} + \frac{1880}{7839} a^{3} + \frac{14269}{39195} a^{2} + \frac{7597}{39195} a - \frac{31}{7839}$, $\frac{1}{509535} a^{16} + \frac{2}{169845} a^{15} - \frac{16103}{509535} a^{14} + \frac{922}{101907} a^{13} + \frac{1993}{509535} a^{12} - \frac{1361}{509535} a^{11} - \frac{337}{13065} a^{10} + \frac{5336}{169845} a^{9} - \frac{44072}{101907} a^{8} + \frac{164054}{509535} a^{7} + \frac{243326}{509535} a^{6} + \frac{8543}{56615} a^{5} - \frac{125072}{509535} a^{4} + \frac{31051}{509535} a^{3} + \frac{210376}{509535} a^{2} + \frac{153799}{509535} a + \frac{3343}{11323}$, $\frac{1}{100231139385} a^{17} + \frac{58531}{100231139385} a^{16} - \frac{60167}{20046227877} a^{15} - \frac{128430734}{33410379795} a^{14} - \frac{572319479}{100231139385} a^{13} - \frac{208214699}{7710087645} a^{12} - \frac{1542068662}{100231139385} a^{11} + \frac{6762554752}{100231139385} a^{10} + \frac{5438563193}{100231139385} a^{9} - \frac{50061731807}{100231139385} a^{8} + \frac{1165517608}{33410379795} a^{7} - \frac{244075822}{1542017529} a^{6} - \frac{2825868458}{33410379795} a^{5} - \frac{30307864594}{100231139385} a^{4} + \frac{2923921984}{11136793265} a^{3} + \frac{702217442}{33410379795} a^{2} - \frac{19312396282}{100231139385} a - \frac{2444136544}{20046227877}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{64259656}{11136793265} a^{17} + \frac{1483980817}{33410379795} a^{16} - \frac{1308279124}{6682075959} a^{15} + \frac{3736208222}{6682075959} a^{14} - \frac{38451552422}{33410379795} a^{13} + \frac{59936437858}{33410379795} a^{12} - \frac{71673079382}{33410379795} a^{11} + \frac{68024457164}{33410379795} a^{10} - \frac{100318186318}{33410379795} a^{9} + \frac{227802295316}{33410379795} a^{8} - \frac{10358995994}{856676405} a^{7} + \frac{29450285438}{2227358653} a^{6} - \frac{70620724062}{11136793265} a^{5} - \frac{39140535223}{6682075959} a^{4} + \frac{419780678266}{33410379795} a^{3} - \frac{23999553368}{2570029215} a^{2} + \frac{1615832307}{856676405} a + \frac{564780243}{2227358653} \) (order $6$)
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:  \( 61890.554687 \)
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

\(\Q(\sqrt{-3}) \), 3.1.4563.1 x3, 3.3.169.1, 6.0.62462907.1, 6.0.369603.2 x2, 6.0.771147.1, 9.3.95006081547.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.369603.2
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$