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Properties

Label 18.0.66513197260...1875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 5^{12}\cdot 7^{12}$
Root discriminant $18.53$
Ramified primes $3, 5, 7$
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![1, -3, 13, -38, 35, 87, -215, -61, 935, -1903, 2255, -1909, 1291, -735, 353, -140, 43, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 43*x^16 - 140*x^15 + 353*x^14 - 735*x^13 + 1291*x^12 - 1909*x^11 + 2255*x^10 - 1903*x^9 + 935*x^8 - 61*x^7 - 215*x^6 + 87*x^5 + 35*x^4 - 38*x^3 + 13*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^18 - 9*x^17 + 43*x^16 - 140*x^15 + 353*x^14 - 735*x^13 + 1291*x^12 - 1909*x^11 + 2255*x^10 - 1903*x^9 + 935*x^8 - 61*x^7 - 215*x^6 + 87*x^5 + 35*x^4 - 38*x^3 + 13*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 43 x^{16} - 140 x^{15} + 353 x^{14} - 735 x^{13} + 1291 x^{12} - 1909 x^{11} + 2255 x^{10} - 1903 x^{9} + 935 x^{8} - 61 x^{7} - 215 x^{6} + 87 x^{5} + 35 x^{4} - 38 x^{3} + 13 x^{2} - 3 x + 1 \)

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:  \(-66513197260078857421875=-\,3^{9}\cdot 5^{12}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{606429} a^{16} - \frac{8}{606429} a^{15} + \frac{65032}{606429} a^{14} + \frac{151345}{606429} a^{13} + \frac{226472}{606429} a^{12} - \frac{294536}{606429} a^{11} - \frac{47926}{202143} a^{10} + \frac{255022}{606429} a^{9} + \frac{1469}{4701} a^{8} - \frac{277754}{606429} a^{7} + \frac{48896}{202143} a^{6} - \frac{245210}{606429} a^{5} - \frac{242227}{606429} a^{4} + \frac{10468}{606429} a^{3} + \frac{91882}{606429} a^{2} + \frac{67102}{606429} a + \frac{8677}{606429}$, $\frac{1}{43056459} a^{17} + \frac{3}{4784051} a^{16} - \frac{711536}{4784051} a^{15} - \frac{2357752}{14352153} a^{14} - \frac{16307897}{43056459} a^{13} - \frac{207994}{1001313} a^{12} + \frac{20475341}{43056459} a^{11} - \frac{532205}{43056459} a^{10} - \frac{21408322}{43056459} a^{9} - \frac{9412373}{43056459} a^{8} + \frac{734591}{43056459} a^{7} - \frac{10676141}{43056459} a^{6} - \frac{17921012}{43056459} a^{5} + \frac{18215399}{43056459} a^{4} - \frac{1397009}{14352153} a^{3} - \frac{4767823}{14352153} a^{2} - \frac{703878}{4784051} a - \frac{15059173}{43056459}$

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:  \( -\frac{725234}{1001313} a^{17} + \frac{6164489}{1001313} a^{16} - \frac{28103350}{1001313} a^{15} + \frac{87485345}{1001313} a^{14} - \frac{70783057}{333771} a^{13} + \frac{47487505}{111257} a^{12} - \frac{724524071}{1001313} a^{11} + \frac{1027351171}{1001313} a^{10} - \frac{1132780187}{1001313} a^{9} + \frac{833635165}{1001313} a^{8} - \frac{291092267}{1001313} a^{7} - \frac{65049254}{1001313} a^{6} + \frac{10483740}{111257} a^{5} - \frac{833821}{111257} a^{4} - \frac{25313632}{1001313} a^{3} + \frac{13333118}{1001313} a^{2} - \frac{4626937}{1001313} a + \frac{1679656}{1001313} \) (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:  \( 21968.4785291 \)
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

\(\Q(\sqrt{-3}) \), 3.1.3675.1 x3, \(\Q(\zeta_{7})^+\), 6.0.40516875.1, 6.0.826875.2 x2, 6.0.64827.1, 9.3.49633171875.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.826875.2
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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$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$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.7.3t1.1c1$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3_7.6t1.2c1$1$ $ 3 \cdot 7 $ $x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$ $C_6$ (as 6T1) $0$ $-1$
* 1.3_7.6t1.2c2$1$ $ 3 \cdot 7 $ $x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$ $C_6$ (as 6T1) $0$ $-1$
* 1.7.3t1.1c2$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
*2 2.3_5e2_7e2.3t2.1c1$2$ $ 3 \cdot 5^{2} \cdot 7^{2}$ $x^{3} - 35$ $S_3$ (as 3T2) $1$ $0$
*2 2.3_5e2_7.6t5.3c1$2$ $ 3 \cdot 5^{2} \cdot 7 $ $x^{18} - 9 x^{17} + 43 x^{16} - 140 x^{15} + 353 x^{14} - 735 x^{13} + 1291 x^{12} - 1909 x^{11} + 2255 x^{10} - 1903 x^{9} + 935 x^{8} - 61 x^{7} - 215 x^{6} + 87 x^{5} + 35 x^{4} - 38 x^{3} + 13 x^{2} - 3 x + 1$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.3_5e2_7.6t5.3c2$2$ $ 3 \cdot 5^{2} \cdot 7 $ $x^{18} - 9 x^{17} + 43 x^{16} - 140 x^{15} + 353 x^{14} - 735 x^{13} + 1291 x^{12} - 1909 x^{11} + 2255 x^{10} - 1903 x^{9} + 935 x^{8} - 61 x^{7} - 215 x^{6} + 87 x^{5} + 35 x^{4} - 38 x^{3} + 13 x^{2} - 3 x + 1$ $S_3 \times C_3$ (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 *.