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Properties

Label 18.0.16839991231...7632.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{9}\cdot 7^{12}\cdot 11^{9}$
Root discriminant $42.04$
Ramified primes $2, 3, 7, 11$
Class number $324$ (GRH)
Class group $[3, 6, 18]$ (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![74088, -127008, 372456, -241380, 549366, -223626, 442963, -125158, 205793, -43522, 56719, -9216, 9377, -1134, 907, -74, 47, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 47*x^16 - 74*x^15 + 907*x^14 - 1134*x^13 + 9377*x^12 - 9216*x^11 + 56719*x^10 - 43522*x^9 + 205793*x^8 - 125158*x^7 + 442963*x^6 - 223626*x^5 + 549366*x^4 - 241380*x^3 + 372456*x^2 - 127008*x + 74088)
 
gp: K = bnfinit(x^18 - 2*x^17 + 47*x^16 - 74*x^15 + 907*x^14 - 1134*x^13 + 9377*x^12 - 9216*x^11 + 56719*x^10 - 43522*x^9 + 205793*x^8 - 125158*x^7 + 442963*x^6 - 223626*x^5 + 549366*x^4 - 241380*x^3 + 372456*x^2 - 127008*x + 74088, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 47 x^{16} - 74 x^{15} + 907 x^{14} - 1134 x^{13} + 9377 x^{12} - 9216 x^{11} + 56719 x^{10} - 43522 x^{9} + 205793 x^{8} - 125158 x^{7} + 442963 x^{6} - 223626 x^{5} + 549366 x^{4} - 241380 x^{3} + 372456 x^{2} - 127008 x + 74088 \)

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:  \(-168399912313071389456086597632=-\,2^{18}\cdot 3^{9}\cdot 7^{12}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.04$
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:  $2, 3, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{246} a^{14} + \frac{47}{123} a^{13} - \frac{85}{246} a^{12} - \frac{7}{123} a^{11} + \frac{73}{246} a^{10} - \frac{10}{41} a^{9} - \frac{121}{246} a^{8} + \frac{18}{41} a^{7} - \frac{29}{246} a^{6} + \frac{49}{123} a^{5} + \frac{23}{246} a^{4} - \frac{20}{123} a^{3} - \frac{65}{246} a^{2} - \frac{10}{41} a + \frac{7}{41}$, $\frac{1}{246} a^{15} - \frac{65}{246} a^{13} + \frac{52}{123} a^{12} - \frac{29}{82} a^{11} - \frac{17}{123} a^{10} + \frac{107}{246} a^{9} - \frac{40}{123} a^{8} - \frac{95}{246} a^{7} + \frac{59}{123} a^{6} - \frac{29}{82} a^{5} + \frac{2}{41} a^{4} + \frac{5}{246} a^{3} - \frac{50}{123} a^{2} + \frac{4}{41} a - \frac{2}{41}$, $\frac{1}{1748370708} a^{16} - \frac{48773}{437092677} a^{15} + \frac{669107}{1748370708} a^{14} - \frac{17643305}{437092677} a^{13} + \frac{93437383}{1748370708} a^{12} + \frac{2994306}{6937979} a^{11} - \frac{596208847}{1748370708} a^{10} - \frac{112900679}{291395118} a^{9} + \frac{289142887}{1748370708} a^{8} - \frac{181361242}{437092677} a^{7} - \frac{78651865}{249767244} a^{6} + \frac{78486713}{437092677} a^{5} + \frac{473488795}{1748370708} a^{4} + \frac{46952992}{145697559} a^{3} - \frac{122481799}{291395118} a^{2} - \frac{6261548}{48565853} a + \frac{875578}{6937979}$, $\frac{1}{1404974762980899931379231316} a^{17} + \frac{35778097738876855}{1404974762980899931379231316} a^{16} - \frac{693725954695175894710507}{1404974762980899931379231316} a^{15} + \frac{1194808453882077372859657}{1404974762980899931379231316} a^{14} - \frac{158202162639395095706128703}{1404974762980899931379231316} a^{13} - \frac{9466512850388455782192863}{66903560141947615779963396} a^{12} + \frac{426030260579006028241140611}{1404974762980899931379231316} a^{11} - \frac{5453613810136625420373151}{156108306997877770153247924} a^{10} + \frac{465414928315401339761361559}{1404974762980899931379231316} a^{9} - \frac{103360718851890007989136939}{1404974762980899931379231316} a^{8} - \frac{52358267114071972358507071}{200710680425842847339890188} a^{7} - \frac{308146875800501225551786945}{1404974762980899931379231316} a^{6} - \frac{649576805067640927024804715}{1404974762980899931379231316} a^{5} + \frac{149192265258790085262698419}{468324920993633310459743772} a^{4} - \frac{38365154170020901799579237}{117081230248408327614935943} a^{3} - \frac{26171678530177086001441193}{234162460496816655229871886} a^{2} + \frac{406022296783908139163128}{5575296678495634648330283} a - \frac{153139892175041023902970}{796470954070804949761469}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{18}$, which has order $324$ (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:  \( 130703.898439 \) (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

\(\Q(\sqrt{-33}) \), 3.1.6468.1 x3, \(\Q(\zeta_{7})^+\), 6.0.5522223168.1, 6.0.112698432.1 x2, 6.0.5522223168.10, 9.3.270588935232.2 x3

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

Sibling fields

Degree 6 sibling: 6.0.112698432.1
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.6.3$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$$[2]^{3}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$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$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$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_3_11.2t1.1c1$1$ $ 2^{2} \cdot 3 \cdot 11 $ $x^{2} + 33$ $C_2$ (as 2T1) $1$ $-1$
* 1.2e2_3_7_11.6t1.1c1$1$ $ 2^{2} \cdot 3 \cdot 7 \cdot 11 $ $x^{6} - 2 x^{5} + 96 x^{4} - 126 x^{3} + 3335 x^{2} - 2248 x + 41581$ $C_6$ (as 6T1) $0$ $-1$
* 1.2e2_3_7_11.6t1.1c2$1$ $ 2^{2} \cdot 3 \cdot 7 \cdot 11 $ $x^{6} - 2 x^{5} + 96 x^{4} - 126 x^{3} + 3335 x^{2} - 2248 x + 41581$ $C_6$ (as 6T1) $0$ $-1$
* 1.7.3t1.1c1$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.7.3t1.1c2$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
*2 2.2e2_3_7e2_11.3t2.1c1$2$ $ 2^{2} \cdot 3 \cdot 7^{2} \cdot 11 $ $x^{3} - x^{2} + 12 x - 6$ $S_3$ (as 3T2) $1$ $0$
*2 2.2e2_3_7_11.6t5.3c1$2$ $ 2^{2} \cdot 3 \cdot 7 \cdot 11 $ $x^{18} - 2 x^{17} + 47 x^{16} - 74 x^{15} + 907 x^{14} - 1134 x^{13} + 9377 x^{12} - 9216 x^{11} + 56719 x^{10} - 43522 x^{9} + 205793 x^{8} - 125158 x^{7} + 442963 x^{6} - 223626 x^{5} + 549366 x^{4} - 241380 x^{3} + 372456 x^{2} - 127008 x + 74088$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.2e2_3_7_11.6t5.3c2$2$ $ 2^{2} \cdot 3 \cdot 7 \cdot 11 $ $x^{18} - 2 x^{17} + 47 x^{16} - 74 x^{15} + 907 x^{14} - 1134 x^{13} + 9377 x^{12} - 9216 x^{11} + 56719 x^{10} - 43522 x^{9} + 205793 x^{8} - 125158 x^{7} + 442963 x^{6} - 223626 x^{5} + 549366 x^{4} - 241380 x^{3} + 372456 x^{2} - 127008 x + 74088$ $S_3 \times C_3$ (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 *.