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Properties

Label 18.0.29509806056...1568.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{16}\cdot 3^{37}$
Root discriminant $17.71$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_2\times He_3:C_2$ (as 18T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12, 0, -18, 0, 27, -27, 45, -54, -9, 81, -81, 36, 21, -81, 108, -81, 36, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 36*x^16 - 81*x^15 + 108*x^14 - 81*x^13 + 21*x^12 + 36*x^11 - 81*x^10 + 81*x^9 - 9*x^8 - 54*x^7 + 45*x^6 - 27*x^5 + 27*x^4 - 18*x^2 + 12)
 
gp: K = bnfinit(x^18 - 9*x^17 + 36*x^16 - 81*x^15 + 108*x^14 - 81*x^13 + 21*x^12 + 36*x^11 - 81*x^10 + 81*x^9 - 9*x^8 - 54*x^7 + 45*x^6 - 27*x^5 + 27*x^4 - 18*x^2 + 12, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 36 x^{16} - 81 x^{15} + 108 x^{14} - 81 x^{13} + 21 x^{12} + 36 x^{11} - 81 x^{10} + 81 x^{9} - 9 x^{8} - 54 x^{7} + 45 x^{6} - 27 x^{5} + 27 x^{4} - 18 x^{2} + 12 \)

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:  \(-29509806056472403181568=-\,2^{16}\cdot 3^{37}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.71$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{14} a^{15} - \frac{1}{7} a^{14} - \frac{1}{14} a^{13} + \frac{1}{7} a^{12} + \frac{1}{14} a^{11} - \frac{3}{14} a^{10} - \frac{5}{14} a^{9} - \frac{5}{14} a^{8} - \frac{1}{2} a^{7} - \frac{3}{14} a^{6} + \frac{1}{7} a^{5} + \frac{1}{14} a^{4} - \frac{1}{2} a^{3} - \frac{2}{7} a^{2} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{14} a^{16} + \frac{1}{7} a^{14} - \frac{1}{7} a^{12} - \frac{1}{14} a^{11} + \frac{3}{14} a^{10} + \frac{3}{7} a^{9} - \frac{3}{14} a^{8} + \frac{2}{7} a^{7} + \frac{3}{14} a^{6} + \frac{5}{14} a^{5} - \frac{5}{14} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{246249598} a^{17} - \frac{2122514}{123124799} a^{16} - \frac{2229723}{123124799} a^{15} - \frac{4846857}{246249598} a^{14} + \frac{1858155}{246249598} a^{13} - \frac{2582635}{17589257} a^{12} + \frac{2268362}{17589257} a^{11} + \frac{38148615}{246249598} a^{10} - \frac{7522672}{123124799} a^{9} + \frac{17836456}{123124799} a^{8} - \frac{50941565}{123124799} a^{7} - \frac{9717401}{35178514} a^{6} - \frac{51286323}{246249598} a^{5} + \frac{17636235}{123124799} a^{4} - \frac{96548947}{246249598} a^{3} - \frac{87903967}{246249598} a^{2} + \frac{25567726}{123124799} a - \frac{4929159}{123124799}$

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:  \( -\frac{12798042}{123124799} a^{17} + \frac{234667809}{246249598} a^{16} - \frac{935410209}{246249598} a^{15} + \frac{1025894700}{123124799} a^{14} - \frac{2585443743}{246249598} a^{13} + \frac{125291934}{17589257} a^{12} - \frac{51307845}{35178514} a^{11} - \frac{901914723}{246249598} a^{10} + \frac{1049020079}{123124799} a^{9} - \frac{1838303739}{246249598} a^{8} - \frac{103668183}{246249598} a^{7} + \frac{21975057}{5025502} a^{6} - \frac{279140058}{123124799} a^{5} + \frac{724243815}{246249598} a^{4} - \frac{647340231}{246249598} a^{3} - \frac{222118479}{246249598} a^{2} + \frac{95954895}{123124799} a + \frac{201279664}{123124799} \) (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:  \( 41849.8998243 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.324.1, 6.0.314928.2, 9.1.99179645184.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
3Data not computed