↑

Properties

Label 18.0.36566016894...3024.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{9}\cdot 7^{12}$
Root discriminant $17.93$
Ramified primes $2, 3, 7$
Class number $2$
Class group $[2]$
Galois group $S_3 \times C_3$ (as 18T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![216, 0, 540, 0, 66, 0, 337, 0, 292, 0, -19, 0, -30, 0, -6, 0, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 2*x^16 - 6*x^14 - 30*x^12 - 19*x^10 + 292*x^8 + 337*x^6 + 66*x^4 + 540*x^2 + 216)
 
gp: K = bnfinit(x^18 + 2*x^16 - 6*x^14 - 30*x^12 - 19*x^10 + 292*x^8 + 337*x^6 + 66*x^4 + 540*x^2 + 216, 1)
 

Normalized defining polynomial

\( x^{18} + 2 x^{16} - 6 x^{14} - 30 x^{12} - 19 x^{10} + 292 x^{8} + 337 x^{6} + 66 x^{4} + 540 x^{2} + 216 \)

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:  \(-36566016894007743873024=-\,2^{27}\cdot 3^{9}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.93$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{14} + \frac{1}{12} a^{12} - \frac{1}{2} a^{7} + \frac{11}{24} a^{6} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{5}{24} a^{2} - \frac{1}{4}$, $\frac{1}{24} a^{15} + \frac{1}{12} a^{13} + \frac{11}{24} a^{7} - \frac{1}{3} a^{5} - \frac{5}{24} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{68214811968} a^{16} + \frac{234384785}{17053702992} a^{14} - \frac{175177543}{3789711776} a^{12} - \frac{1231538303}{11369135328} a^{10} - \frac{460054135}{68214811968} a^{8} - \frac{7075193845}{34107405984} a^{6} - \frac{1}{2} a^{5} + \frac{23025577165}{68214811968} a^{4} - \frac{1}{2} a^{3} - \frac{854676391}{5684567664} a^{2} - \frac{619456331}{1894855888}$, $\frac{1}{68214811968} a^{17} + \frac{234384785}{17053702992} a^{15} - \frac{175177543}{3789711776} a^{13} - \frac{1231538303}{11369135328} a^{11} - \frac{460054135}{68214811968} a^{9} - \frac{7075193845}{34107405984} a^{7} - \frac{1}{2} a^{6} + \frac{23025577165}{68214811968} a^{5} - \frac{1}{2} a^{4} - \frac{854676391}{5684567664} a^{3} - \frac{619456331}{1894855888} a$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( 2614.29263124 \)
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

\(\Q(\sqrt{-6}) \), 3.1.1176.1 x3, \(\Q(\zeta_{7})^+\), 6.0.33191424.2, 6.0.33191424.1, 6.0.677376.1 x2, 9.3.1626379776.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.677376.1
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 R R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e3_3.2t1.2c1$1$ $ 2^{3} \cdot 3 $ $x^{2} + 6$ $C_2$ (as 2T1) $1$ $-1$
* 1.7.3t1.1c1$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.2e3_3_7.6t1.3c1$1$ $ 2^{3} \cdot 3 \cdot 7 $ $x^{6} - 2 x^{5} + 15 x^{4} - 18 x^{3} + 122 x^{2} - 88 x + 433$ $C_6$ (as 6T1) $0$ $-1$
* 1.7.3t1.1c2$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.2e3_3_7.6t1.3c2$1$ $ 2^{3} \cdot 3 \cdot 7 $ $x^{6} - 2 x^{5} + 15 x^{4} - 18 x^{3} + 122 x^{2} - 88 x + 433$ $C_6$ (as 6T1) $0$ $-1$
*2 2.2e3_3_7e2.3t2.1c1$2$ $ 2^{3} \cdot 3 \cdot 7^{2}$ $x^{3} - x^{2} - 2 x - 6$ $S_3$ (as 3T2) $1$ $0$
*2 2.2e3_3_7.6t5.3c1$2$ $ 2^{3} \cdot 3 \cdot 7 $ $x^{18} + 2 x^{16} - 6 x^{14} - 30 x^{12} - 19 x^{10} + 292 x^{8} + 337 x^{6} + 66 x^{4} + 540 x^{2} + 216$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.2e3_3_7.6t5.3c2$2$ $ 2^{3} \cdot 3 \cdot 7 $ $x^{18} + 2 x^{16} - 6 x^{14} - 30 x^{12} - 19 x^{10} + 292 x^{8} + 337 x^{6} + 66 x^{4} + 540 x^{2} + 216$ $S_3 \times C_3$ (as 18T3) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.