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Properties

Label 18.0.85087260843...9375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 5^{9}\cdot 19^{12}$
Root discriminant $27.58$
Ramified primes $3, 5, 19$
Class number $2$
Class group $[2]$
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![4096, 0, -5440, 5080, -284, -2244, 4993, -6252, 3521, -1680, 2173, -2252, 1361, -536, 165, -64, 29, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 29*x^16 - 64*x^15 + 165*x^14 - 536*x^13 + 1361*x^12 - 2252*x^11 + 2173*x^10 - 1680*x^9 + 3521*x^8 - 6252*x^7 + 4993*x^6 - 2244*x^5 - 284*x^4 + 5080*x^3 - 5440*x^2 + 4096)
 
gp: K = bnfinit(x^18 - 8*x^17 + 29*x^16 - 64*x^15 + 165*x^14 - 536*x^13 + 1361*x^12 - 2252*x^11 + 2173*x^10 - 1680*x^9 + 3521*x^8 - 6252*x^7 + 4993*x^6 - 2244*x^5 - 284*x^4 + 5080*x^3 - 5440*x^2 + 4096, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} + 29 x^{16} - 64 x^{15} + 165 x^{14} - 536 x^{13} + 1361 x^{12} - 2252 x^{11} + 2173 x^{10} - 1680 x^{9} + 3521 x^{8} - 6252 x^{7} + 4993 x^{6} - 2244 x^{5} - 284 x^{4} + 5080 x^{3} - 5440 x^{2} + 4096 \)

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:  \(-85087260843709466724609375=-\,3^{9}\cdot 5^{9}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.58$
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:  $3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{308} a^{14} + \frac{37}{154} a^{13} + \frac{69}{308} a^{12} + \frac{3}{77} a^{11} + \frac{9}{44} a^{10} + \frac{16}{77} a^{9} - \frac{65}{308} a^{8} + \frac{3}{14} a^{7} - \frac{13}{44} a^{6} + \frac{75}{154} a^{5} + \frac{79}{308} a^{4} - \frac{3}{77} a^{3} + \frac{83}{308} a^{2} - \frac{5}{77} a + \frac{12}{77}$, $\frac{1}{616} a^{15} + \frac{137}{616} a^{13} - \frac{3}{154} a^{12} - \frac{5}{56} a^{11} - \frac{3}{14} a^{10} - \frac{27}{616} a^{9} - \frac{13}{154} a^{8} - \frac{47}{616} a^{7} - \frac{25}{77} a^{6} - \frac{87}{616} a^{5} + \frac{37}{154} a^{4} + \frac{201}{616} a^{3} - \frac{39}{154} a^{2} + \frac{37}{77} a + \frac{18}{77}$, $\frac{1}{4928} a^{16} - \frac{3}{4928} a^{14} - \frac{103}{616} a^{13} - \frac{475}{4928} a^{12} - \frac{17}{308} a^{11} + \frac{1009}{4928} a^{10} - \frac{97}{1232} a^{9} - \frac{73}{448} a^{8} - \frac{25}{616} a^{7} - \frac{1823}{4928} a^{6} + \frac{23}{1232} a^{5} + \frac{1153}{4928} a^{4} + \frac{73}{1232} a^{3} - \frac{367}{1232} a^{2} - \frac{17}{616} a - \frac{4}{11}$, $\frac{1}{1680174733812672413681795584} a^{17} - \frac{690458605538249939345}{30003120246654864530032064} a^{16} + \frac{1150806273229639086549661}{1680174733812672413681795584} a^{15} - \frac{29363631571574935874611}{105010920863292025855112224} a^{14} - \frac{3716568696580495359637083}{1680174733812672413681795584} a^{13} + \frac{45633532288574934671007311}{210021841726584051710224448} a^{12} - \frac{47188937759235666267831599}{1680174733812672413681795584} a^{11} + \frac{54847337899615930577313441}{420043683453168103420448896} a^{10} + \frac{305320095424235695980242365}{1680174733812672413681795584} a^{9} - \frac{747892849129438796052365}{13126365107911503231889028} a^{8} + \frac{93460875863276902678977985}{1680174733812672413681795584} a^{7} - \frac{15968068885776513657446389}{38185789404833463947313536} a^{6} - \frac{6301205670952442018990267}{21820451088476265112750592} a^{5} - \frac{155766679475720319213810973}{420043683453168103420448896} a^{4} + \frac{14197415376242500752674587}{38185789404833463947313536} a^{3} - \frac{5382138158397482755334109}{210021841726584051710224448} a^{2} + \frac{5636468685320357124968661}{26252730215823006463778056} a + \frac{847356116974511742031347}{3281591276977875807972257}$

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:  \( 221984.4857 \)
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

\(\Q(\sqrt{-15}) \), 3.1.5415.1 x3, 3.3.361.1, 6.0.439833375.3, 6.0.1218375.1 x2, 6.0.439833375.1, 9.3.158779848375.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.1218375.1
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$3.6.3.1$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$$[\ ]_{2}^{3}$
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$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}$
$19$19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$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_5.2t1.1c1$1$ $ 3 \cdot 5 $ $x^{2} - x + 4$ $C_2$ (as 2T1) $1$ $-1$
* 1.3_5_19.6t1.2c1$1$ $ 3 \cdot 5 \cdot 19 $ $x^{6} - x^{5} - x^{4} - 9 x^{3} + 87 x^{2} + 145 x + 379$ $C_6$ (as 6T1) $0$ $-1$
* 1.3_5_19.6t1.2c2$1$ $ 3 \cdot 5 \cdot 19 $ $x^{6} - x^{5} - x^{4} - 9 x^{3} + 87 x^{2} + 145 x + 379$ $C_6$ (as 6T1) $0$ $-1$
* 1.19.3t1.1c1$1$ $ 19 $ $x^{3} - x^{2} - 6 x + 7$ $C_3$ (as 3T1) $0$ $1$
* 1.19.3t1.1c2$1$ $ 19 $ $x^{3} - x^{2} - 6 x + 7$ $C_3$ (as 3T1) $0$ $1$
*2 2.3_5_19e2.3t2.1c1$2$ $ 3 \cdot 5 \cdot 19^{2}$ $x^{3} - x^{2} - 6 x + 45$ $S_3$ (as 3T2) $1$ $0$
*2 2.3_5_19.6t5.3c1$2$ $ 3 \cdot 5 \cdot 19 $ $x^{18} - 8 x^{17} + 29 x^{16} - 64 x^{15} + 165 x^{14} - 536 x^{13} + 1361 x^{12} - 2252 x^{11} + 2173 x^{10} - 1680 x^{9} + 3521 x^{8} - 6252 x^{7} + 4993 x^{6} - 2244 x^{5} - 284 x^{4} + 5080 x^{3} - 5440 x^{2} + 4096$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.3_5_19.6t5.3c2$2$ $ 3 \cdot 5 \cdot 19 $ $x^{18} - 8 x^{17} + 29 x^{16} - 64 x^{15} + 165 x^{14} - 536 x^{13} + 1361 x^{12} - 2252 x^{11} + 2173 x^{10} - 1680 x^{9} + 3521 x^{8} - 6252 x^{7} + 4993 x^{6} - 2244 x^{5} - 284 x^{4} + 5080 x^{3} - 5440 x^{2} + 4096$ $S_3 \times C_3$ (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 *.