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Properties

Label 18.0.27243805597...0000.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 5^{12}\cdot 7^{12}$
Root discriminant $29.42$
Ramified primes $2, 3, 5, 7$
Class number $3$
Class group $[3]$
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36, -144, -48, -1164, 7654, -7788, 2837, -1405, 1977, -1568, 873, -417, 204, -125, 69, -22, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 5*x^16 - 22*x^15 + 69*x^14 - 125*x^13 + 204*x^12 - 417*x^11 + 873*x^10 - 1568*x^9 + 1977*x^8 - 1405*x^7 + 2837*x^6 - 7788*x^5 + 7654*x^4 - 1164*x^3 - 48*x^2 - 144*x + 36)
 
gp: K = bnfinit(x^18 - 3*x^17 + 5*x^16 - 22*x^15 + 69*x^14 - 125*x^13 + 204*x^12 - 417*x^11 + 873*x^10 - 1568*x^9 + 1977*x^8 - 1405*x^7 + 2837*x^6 - 7788*x^5 + 7654*x^4 - 1164*x^3 - 48*x^2 - 144*x + 36, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 5 x^{16} - 22 x^{15} + 69 x^{14} - 125 x^{13} + 204 x^{12} - 417 x^{11} + 873 x^{10} - 1568 x^{9} + 1977 x^{8} - 1405 x^{7} + 2837 x^{6} - 7788 x^{5} + 7654 x^{4} - 1164 x^{3} - 48 x^{2} - 144 x + 36 \)

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:  \(-272438055977283000000000000=-\,2^{12}\cdot 3^{9}\cdot 5^{12}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.42$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{5}$, $\frac{1}{54} a^{12} + \frac{1}{27} a^{11} - \frac{1}{27} a^{10} - \frac{1}{18} a^{9} + \frac{1}{9} a^{8} - \frac{1}{54} a^{6} - \frac{1}{27} a^{5} + \frac{10}{27} a^{4} + \frac{1}{18} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3}$, $\frac{1}{54} a^{13} + \frac{1}{54} a^{10} + \frac{1}{9} a^{8} - \frac{1}{54} a^{7} + \frac{17}{54} a^{4} + \frac{2}{9} a^{2} + \frac{1}{3}$, $\frac{1}{324} a^{14} + \frac{1}{162} a^{13} - \frac{5}{324} a^{11} + \frac{1}{162} a^{10} - \frac{43}{324} a^{8} - \frac{19}{162} a^{7} - \frac{1}{9} a^{6} - \frac{157}{324} a^{5} - \frac{55}{162} a^{4} + \frac{1}{3} a^{3} + \frac{7}{54} a^{2} - \frac{2}{9} a + \frac{4}{9}$, $\frac{1}{972} a^{15} + \frac{1}{486} a^{13} - \frac{5}{972} a^{12} + \frac{4}{81} a^{11} + \frac{1}{486} a^{10} - \frac{43}{972} a^{9} - \frac{2}{81} a^{8} - \frac{37}{486} a^{7} + \frac{23}{972} a^{6} - \frac{22}{81} a^{5} + \frac{107}{486} a^{4} - \frac{47}{162} a^{3} + \frac{38}{81} a^{2} - \frac{10}{27} a - \frac{5}{27}$, $\frac{1}{235224} a^{16} + \frac{95}{235224} a^{15} + \frac{37}{117612} a^{14} - \frac{1885}{235224} a^{13} + \frac{527}{235224} a^{12} - \frac{5153}{117612} a^{11} + \frac{4303}{78408} a^{10} - \frac{1139}{235224} a^{9} + \frac{3863}{117612} a^{8} - \frac{18869}{235224} a^{7} + \frac{2263}{235224} a^{6} + \frac{33071}{117612} a^{5} - \frac{28109}{117612} a^{4} - \frac{1669}{6534} a^{3} + \frac{422}{9801} a^{2} + \frac{58}{3267} a - \frac{2221}{6534}$, $\frac{1}{1245703728456} a^{17} + \frac{405827}{311425932114} a^{16} + \frac{31024321}{415234576152} a^{15} - \frac{152754853}{1245703728456} a^{14} + \frac{688199147}{622851864228} a^{13} + \frac{3193652749}{415234576152} a^{12} + \frac{14391917831}{415234576152} a^{11} - \frac{1885291235}{207617288076} a^{10} - \frac{8123234341}{415234576152} a^{9} + \frac{8595908287}{1245703728456} a^{8} - \frac{64432407581}{622851864228} a^{7} - \frac{42956607355}{415234576152} a^{6} - \frac{205887487193}{622851864228} a^{5} - \frac{67497975307}{622851864228} a^{4} - \frac{13262172139}{34602881346} a^{3} + \frac{1888394191}{4513419306} a^{2} + \frac{647455883}{3145716486} a - \frac{11813699273}{34602881346}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( \frac{80041913}{3431690712} a^{17} - \frac{214977701}{3431690712} a^{16} + \frac{55203773}{571948452} a^{15} - \frac{1651709819}{3431690712} a^{14} + \frac{4996308431}{3431690712} a^{13} - \frac{1400370739}{571948452} a^{12} + \frac{4524989513}{1143896904} a^{11} - \frac{9636851881}{1143896904} a^{10} + \frac{10078465345}{571948452} a^{9} - \frac{105829846579}{3431690712} a^{8} + \frac{123579584359}{3431690712} a^{7} - \frac{11843138939}{571948452} a^{6} + \frac{50363151085}{857922678} a^{5} - \frac{69710401220}{428961339} a^{4} + \frac{12028943263}{95324742} a^{3} + \frac{205439405}{12433662} a^{2} - \frac{1102633}{95324742} a - \frac{116868029}{47662371} \) (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:  \( 25716902.8477 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.300.1 x3, 3.1.14700.1 x3, 3.1.3675.1 x3, 3.1.588.1 x3, 6.0.270000.1, 6.0.648270000.1, 6.0.40516875.1, 6.0.1037232.1, 9.1.9529569000000.1 x9

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

Sibling fields

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 R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$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}$