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Properties

Label 18.0.24238444506...0704.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{24}\cdot 41^{9}$
Root discriminant $55.41$
Ramified primes $2, 3, 41$
Class number $648$ (GRH)
Class group $[3, 3, 72]$ (GRH)
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![5184, 20736, 145152, 59040, 419040, -252432, 339768, -109152, 71424, -1196, 11586, 270, 3095, 150, 315, -28, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 28*x^15 + 315*x^14 + 150*x^13 + 3095*x^12 + 270*x^11 + 11586*x^10 - 1196*x^9 + 71424*x^8 - 109152*x^7 + 339768*x^6 - 252432*x^5 + 419040*x^4 + 59040*x^3 + 145152*x^2 + 20736*x + 5184)
 
gp: K = bnfinit(x^18 - 28*x^15 + 315*x^14 + 150*x^13 + 3095*x^12 + 270*x^11 + 11586*x^10 - 1196*x^9 + 71424*x^8 - 109152*x^7 + 339768*x^6 - 252432*x^5 + 419040*x^4 + 59040*x^3 + 145152*x^2 + 20736*x + 5184, 1)
 

Normalized defining polynomial

\( x^{18} - 28 x^{15} + 315 x^{14} + 150 x^{13} + 3095 x^{12} + 270 x^{11} + 11586 x^{10} - 1196 x^{9} + 71424 x^{8} - 109152 x^{7} + 339768 x^{6} - 252432 x^{5} + 419040 x^{4} + 59040 x^{3} + 145152 x^{2} + 20736 x + 5184 \)

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:  \(-24238444506812135997594990280704=-\,2^{18}\cdot 3^{24}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.41$
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:  $2, 3, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{6} a^{3}$, $\frac{1}{12} a^{10} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{12} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{11} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{12} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{72} a^{12} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{72} a^{6} + \frac{1}{4} a^{5} + \frac{5}{12} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{72} a^{13} + \frac{1}{24} a^{9} - \frac{1}{4} a^{8} - \frac{1}{72} a^{7} + \frac{1}{4} a^{6} - \frac{1}{12} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{288} a^{14} - \frac{1}{24} a^{11} - \frac{1}{32} a^{10} + \frac{1}{48} a^{9} + \frac{71}{288} a^{8} - \frac{1}{16} a^{7} + \frac{17}{48} a^{6} - \frac{11}{24} a^{5} - \frac{3}{8} a^{4} + \frac{5}{12} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{31968} a^{15} + \frac{41}{31968} a^{14} + \frac{1}{333} a^{13} + \frac{1}{444} a^{12} + \frac{9}{1184} a^{11} - \frac{91}{3552} a^{10} + \frac{413}{31968} a^{9} - \frac{2615}{31968} a^{8} + \frac{71}{2664} a^{7} + \frac{1417}{5328} a^{6} + \frac{1}{111} a^{5} + \frac{427}{888} a^{4} + \frac{221}{444} a^{3} - \frac{5}{111} a^{2} + \frac{23}{74} a + \frac{1}{74}$, $\frac{1}{196283520} a^{16} - \frac{169}{32713920} a^{15} + \frac{81151}{98141760} a^{14} + \frac{7225}{1090464} a^{13} + \frac{20789}{65427840} a^{12} + \frac{82877}{5452320} a^{11} + \frac{7580069}{196283520} a^{10} + \frac{68899}{8178480} a^{9} + \frac{2142049}{49070880} a^{8} - \frac{491629}{5452320} a^{7} + \frac{205499}{1363080} a^{6} - \frac{223841}{545232} a^{5} + \frac{35259}{908720} a^{4} - \frac{142259}{454360} a^{3} - \frac{3283}{36840} a^{2} - \frac{28893}{227180} a - \frac{70291}{227180}$, $\frac{1}{143376366735993648560117658998400} a^{17} + \frac{8467602213465118097273}{143376366735993648560117658998400} a^{16} - \frac{36019876619264370386964037}{17922045841999206070014707374800} a^{15} - \frac{17733258279062300376713178743}{71688183367996824280058829499200} a^{14} + \frac{3888117728056590339770883281}{5310235805036801798522876259200} a^{13} + \frac{247396441388715115881149639767}{47792122245331216186705886332800} a^{12} - \frac{3810708246302080470511548851107}{143376366735993648560117658998400} a^{11} + \frac{3459779890658774563969532681359}{143376366735993648560117658998400} a^{10} + \frac{79893936049811873519335205131}{1120127865124950379375919210925} a^{9} - \frac{11958834331932936710853740117}{180121063738685488140851330400} a^{8} + \frac{1485303572756780223215559892699}{11948030561332804046676471583200} a^{7} + \frac{1005245064783237776388099940789}{2987007640333201011669117895800} a^{6} + \frac{114681563830259551542817150471}{995669213444400337223039298600} a^{5} + \frac{2517855442612795526390580227}{398267685377760134889215719440} a^{4} + \frac{855869572753184452568618921}{4978346067222001686115196493} a^{3} - \frac{45644044299070068960130341203}{199133842688880067444607859720} a^{2} + \frac{14894919742928690725445612562}{41486217226850014050959970775} a + \frac{51233293759338825513826845853}{165944868907400056203839883100}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{72}$, which has order $648$ (assuming GRH)

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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2935408.29793 \) (assuming GRH)
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

\(\Q(\sqrt{-41}) \), 3.1.13284.1 x3, \(\Q(\zeta_{9})^+\), 6.0.28940203584.1, 6.0.28940203584.4, 6.0.357286464.1 x2, 9.3.2344156490304.2 x3

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

Sibling fields

Degree 6 sibling: 6.0.357286464.1
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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$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$$[2]$
$41$41.6.3.1$x^{6} - 82 x^{4} + 1681 x^{2} - 11647649$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
41.6.3.1$x^{6} - 82 x^{4} + 1681 x^{2} - 11647649$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
41.6.3.1$x^{6} - 82 x^{4} + 1681 x^{2} - 11647649$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e2_41.2t1.1c1$1$ $ 2^{2} \cdot 41 $ $x^{2} + 41$ $C_2$ (as 2T1) $1$ $-1$
* 1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.2e2_3e2_41.6t1.1c1$1$ $ 2^{2} \cdot 3^{2} \cdot 41 $ $x^{6} + 117 x^{4} - 2 x^{3} + 5052 x^{2} + 252 x + 79377$ $C_6$ (as 6T1) $0$ $-1$
* 1.2e2_3e2_41.6t1.1c2$1$ $ 2^{2} \cdot 3^{2} \cdot 41 $ $x^{6} + 117 x^{4} - 2 x^{3} + 5052 x^{2} + 252 x + 79377$ $C_6$ (as 6T1) $0$ $-1$
* 1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
*2 2.2e2_3e4_41.3t2.1c1$2$ $ 2^{2} \cdot 3^{4} \cdot 41 $ $x^{3} + 6 x - 44$ $S_3$ (as 3T2) $1$ $0$
*2 2.2e2_3e2_41.6t5.3c1$2$ $ 2^{2} \cdot 3^{2} \cdot 41 $ $x^{18} - 28 x^{15} + 315 x^{14} + 150 x^{13} + 3095 x^{12} + 270 x^{11} + 11586 x^{10} - 1196 x^{9} + 71424 x^{8} - 109152 x^{7} + 339768 x^{6} - 252432 x^{5} + 419040 x^{4} + 59040 x^{3} + 145152 x^{2} + 20736 x + 5184$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.2e2_3e2_41.6t5.3c2$2$ $ 2^{2} \cdot 3^{2} \cdot 41 $ $x^{18} - 28 x^{15} + 315 x^{14} + 150 x^{13} + 3095 x^{12} + 270 x^{11} + 11586 x^{10} - 1196 x^{9} + 71424 x^{8} - 109152 x^{7} + 339768 x^{6} - 252432 x^{5} + 419040 x^{4} + 59040 x^{3} + 145152 x^{2} + 20736 x + 5184$ $S_3 \times C_3$ (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 *.