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Properties

Label 18.0.26150883007...0000.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 5^{14}$
Root discriminant $20.00$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37, -201, 459, -679, 918, -1128, 1139, -1104, 999, -724, 552, -456, 263, -123, 84, -46, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 18*x^16 - 46*x^15 + 84*x^14 - 123*x^13 + 263*x^12 - 456*x^11 + 552*x^10 - 724*x^9 + 999*x^8 - 1104*x^7 + 1139*x^6 - 1128*x^5 + 918*x^4 - 679*x^3 + 459*x^2 - 201*x + 37)
 
gp: K = bnfinit(x^18 - 6*x^17 + 18*x^16 - 46*x^15 + 84*x^14 - 123*x^13 + 263*x^12 - 456*x^11 + 552*x^10 - 724*x^9 + 999*x^8 - 1104*x^7 + 1139*x^6 - 1128*x^5 + 918*x^4 - 679*x^3 + 459*x^2 - 201*x + 37, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 18 x^{16} - 46 x^{15} + 84 x^{14} - 123 x^{13} + 263 x^{12} - 456 x^{11} + 552 x^{10} - 724 x^{9} + 999 x^{8} - 1104 x^{7} + 1139 x^{6} - 1128 x^{5} + 918 x^{4} - 679 x^{3} + 459 x^{2} - 201 x + 37 \)

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:  \(-261508830075000000000000=-\,2^{12}\cdot 3^{21}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.00$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{8} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{12} + \frac{1}{3} a^{10} + \frac{4}{9} a^{9} + \frac{4}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{27} a^{16} + \frac{1}{27} a^{15} - \frac{2}{27} a^{13} - \frac{2}{27} a^{12} + \frac{4}{9} a^{11} - \frac{2}{27} a^{10} + \frac{13}{27} a^{9} + \frac{1}{3} a^{8} - \frac{8}{27} a^{7} - \frac{2}{27} a^{6} - \frac{1}{3} a^{5} - \frac{11}{27} a^{4} + \frac{1}{27} a^{3} + \frac{1}{9} a^{2} - \frac{2}{27} a + \frac{10}{27}$, $\frac{1}{5503881418731} a^{17} - \frac{51491488997}{5503881418731} a^{16} + \frac{43774107829}{1834627139577} a^{15} - \frac{348552093116}{5503881418731} a^{14} + \frac{656885796322}{5503881418731} a^{13} - \frac{75711347146}{1834627139577} a^{12} - \frac{1628223612005}{5503881418731} a^{11} + \frac{957326874499}{5503881418731} a^{10} - \frac{212932405814}{1834627139577} a^{9} + \frac{774669806497}{5503881418731} a^{8} + \frac{1384948433398}{5503881418731} a^{7} - \frac{479994535118}{1834627139577} a^{6} + \frac{363643776349}{5503881418731} a^{5} - \frac{2204050806452}{5503881418731} a^{4} - \frac{110908801066}{1834627139577} a^{3} - \frac{997445208296}{5503881418731} a^{2} + \frac{721858320430}{5503881418731} a + \frac{176123828671}{1834627139577}$

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:  \( \frac{339247184}{128472291} a^{17} - \frac{1724724461}{128472291} a^{16} + \frac{4508918951}{128472291} a^{15} - \frac{11394831421}{128472291} a^{14} + \frac{17868539848}{128472291} a^{13} - \frac{24875591431}{128472291} a^{12} + \frac{65784898328}{128472291} a^{11} - \frac{93541299077}{128472291} a^{10} + \frac{98718035636}{128472291} a^{9} - \frac{151959398053}{128472291} a^{8} + \frac{196535163610}{128472291} a^{7} - \frac{188633589364}{128472291} a^{6} + \frac{206912522231}{128472291} a^{5} - \frac{187383345362}{128472291} a^{4} + \frac{132606446681}{128472291} a^{3} - \frac{103389639529}{128472291} a^{2} + \frac{57481179094}{128472291} a - \frac{12321017653}{128472291} \) (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:  \( 127976.41236898548 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.135.1, 3.1.300.1 x3, 6.0.54675.1, 6.0.270000.1, 9.1.295245000000.2 x3

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
2Data not computed
$3$3.6.7.5$x^{6} + 6 x^{2} + 3$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
3.6.7.5$x^{6} + 6 x^{2} + 3$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
3.6.7.5$x^{6} + 6 x^{2} + 3$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$