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Properties

Label 18.2.31637594439...0000.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{26}\cdot 5^{9}\cdot 17^{6}$
Root discriminant $15.65$
Ramified primes $2, 5, 17$
Class number $1$
Class group Trivial
Galois group $C_2\times S_3\wr C_2$ (as 18T63)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, -4, -1, 15, 6, 3, 1, 1, -22, -1, 1, -3, 6, -15, -1, 4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 4*x^16 - x^15 - 15*x^14 + 6*x^13 - 3*x^12 + x^11 - x^10 - 22*x^9 + x^8 + x^7 + 3*x^6 + 6*x^5 + 15*x^4 - x^3 - 4*x^2 - 3*x - 1)
 
gp: K = bnfinit(x^18 - 3*x^17 + 4*x^16 - x^15 - 15*x^14 + 6*x^13 - 3*x^12 + x^11 - x^10 - 22*x^9 + x^8 + x^7 + 3*x^6 + 6*x^5 + 15*x^4 - x^3 - 4*x^2 - 3*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 4 x^{16} - x^{15} - 15 x^{14} + 6 x^{13} - 3 x^{12} + x^{11} - x^{10} - 22 x^{9} + x^{8} + x^{7} + 3 x^{6} + 6 x^{5} + 15 x^{4} - x^{3} - 4 x^{2} - 3 x - 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3163759443968000000000=2^{26}\cdot 5^{9}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.65$
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:  $2, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} + \frac{3}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{2} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{8}$, $\frac{1}{240} a^{14} - \frac{7}{240} a^{13} + \frac{1}{30} a^{12} + \frac{19}{240} a^{11} - \frac{7}{40} a^{10} - \frac{11}{80} a^{9} - \frac{11}{48} a^{8} + \frac{1}{15} a^{7} + \frac{5}{48} a^{6} + \frac{19}{80} a^{5} - \frac{3}{40} a^{4} + \frac{109}{240} a^{3} + \frac{13}{60} a^{2} + \frac{23}{240} a - \frac{91}{240}$, $\frac{1}{240} a^{15} - \frac{11}{240} a^{13} - \frac{1}{16} a^{12} + \frac{1}{240} a^{11} + \frac{11}{80} a^{10} + \frac{11}{60} a^{9} - \frac{13}{80} a^{8} + \frac{17}{240} a^{7} + \frac{41}{120} a^{6} + \frac{17}{80} a^{5} + \frac{103}{240} a^{4} - \frac{11}{48} a^{3} - \frac{21}{80} a^{2} + \frac{1}{6} a - \frac{37}{240}$, $\frac{1}{480} a^{16} - \frac{1}{480} a^{15} - \frac{1}{480} a^{14} - \frac{7}{240} a^{13} - \frac{1}{20} a^{12} - \frac{3}{80} a^{11} - \frac{109}{480} a^{10} + \frac{7}{480} a^{9} + \frac{113}{240} a^{8} + \frac{11}{32} a^{7} - \frac{27}{160} a^{6} + \frac{11}{240} a^{5} - \frac{19}{240} a^{4} + \frac{31}{240} a^{3} - \frac{217}{480} a^{2} + \frac{51}{160} a - \frac{71}{160}$, $\frac{1}{480} a^{17} - \frac{1}{480} a^{14} + \frac{11}{240} a^{13} - \frac{1}{24} a^{12} - \frac{13}{160} a^{11} + \frac{1}{5} a^{10} + \frac{13}{160} a^{9} - \frac{37}{480} a^{8} - \frac{23}{80} a^{7} + \frac{31}{96} a^{6} + \frac{7}{15} a^{5} - \frac{11}{240} a^{4} - \frac{239}{480} a^{3} - \frac{61}{240} a^{2} + \frac{27}{80} a + \frac{239}{480}$

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:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4778.17272788 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3\wr C_2$ (as 18T63):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 18 conjugacy class representatives for $C_2\times S_3\wr C_2$
Character table for $C_2\times S_3\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 9.1.25154560000.1

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

Sibling fields

Degree 12 siblings: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$$[\ ]_{3}^{2}$
2.12.22.60$x^{12} - 84 x^{10} + 444 x^{8} + 32 x^{6} - 272 x^{4} - 320 x^{2} + 64$$6$$2$$22$$D_6$$[3]_{3}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.12.6.1$x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$