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Properties

Label 18.2.26312486298...3088.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{9}\cdot 7^{12}\cdot 11^{9}$
Root discriminant $33.37$
Ramified primes $2, 3, 7, 11$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $S_3^2$ (as 18T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-638, 1254, -1088, 1387, -4546, 8155, -7721, 3413, 577, -2370, 2713, -2355, 1498, -637, 171, -41, 17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 17*x^16 - 41*x^15 + 171*x^14 - 637*x^13 + 1498*x^12 - 2355*x^11 + 2713*x^10 - 2370*x^9 + 577*x^8 + 3413*x^7 - 7721*x^6 + 8155*x^5 - 4546*x^4 + 1387*x^3 - 1088*x^2 + 1254*x - 638)
 
gp: K = bnfinit(x^18 - 6*x^17 + 17*x^16 - 41*x^15 + 171*x^14 - 637*x^13 + 1498*x^12 - 2355*x^11 + 2713*x^10 - 2370*x^9 + 577*x^8 + 3413*x^7 - 7721*x^6 + 8155*x^5 - 4546*x^4 + 1387*x^3 - 1088*x^2 + 1254*x - 638, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 17 x^{16} - 41 x^{15} + 171 x^{14} - 637 x^{13} + 1498 x^{12} - 2355 x^{11} + 2713 x^{10} - 2370 x^{9} + 577 x^{8} + 3413 x^{7} - 7721 x^{6} + 8155 x^{5} - 4546 x^{4} + 1387 x^{3} - 1088 x^{2} + 1254 x - 638 \)

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:  \(2631248629891740460251353088=2^{12}\cdot 3^{9}\cdot 7^{12}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.37$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{12} a^{14} - \frac{1}{6} a^{13} - \frac{1}{4} a^{12} - \frac{1}{12} a^{11} - \frac{1}{3} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{5}{12} a^{6} - \frac{1}{12} a^{5} + \frac{1}{12} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{96} a^{15} + \frac{1}{32} a^{14} + \frac{23}{96} a^{13} + \frac{7}{48} a^{12} + \frac{9}{32} a^{11} - \frac{47}{96} a^{10} + \frac{5}{16} a^{9} - \frac{15}{32} a^{8} + \frac{5}{96} a^{7} - \frac{3}{16} a^{6} - \frac{5}{96} a^{5} + \frac{37}{96} a^{4} - \frac{17}{96} a^{3} - \frac{5}{48} a^{2} + \frac{1}{8} a - \frac{23}{48}$, $\frac{1}{576} a^{16} + \frac{1}{288} a^{15} + \frac{5}{144} a^{14} - \frac{35}{192} a^{13} + \frac{61}{576} a^{12} + \frac{59}{288} a^{11} + \frac{173}{576} a^{10} - \frac{73}{192} a^{9} + \frac{121}{288} a^{8} + \frac{73}{576} a^{7} + \frac{157}{576} a^{6} + \frac{23}{96} a^{5} + \frac{13}{32} a^{4} + \frac{151}{576} a^{3} - \frac{85}{288} a^{2} - \frac{77}{288} a + \frac{71}{288}$, $\frac{1}{381628778994508808884032} a^{17} + \frac{214900374621102930941}{381628778994508808884032} a^{16} + \frac{26287558365226204303}{47703597374313601110504} a^{15} - \frac{2899737071742710659201}{127209592998169602961344} a^{14} - \frac{19463082997453325823269}{95407194748627202221008} a^{13} - \frac{89232338773801375766999}{381628778994508808884032} a^{12} - \frac{75506720397598806334471}{381628778994508808884032} a^{11} + \frac{2845237929008579730785}{21201598833028267160224} a^{10} - \frac{121562349831660961801027}{381628778994508808884032} a^{9} + \frac{52084387695201173331001}{381628778994508808884032} a^{8} + \frac{80634452330937665573867}{190814389497254404442016} a^{7} - \frac{6352506475358726987403}{42403197666056534320448} a^{6} - \frac{6679520475076669078493}{63604796499084801480672} a^{5} - \frac{26624588542171522922069}{381628778994508808884032} a^{4} - \frac{105302152171390623953675}{381628778994508808884032} a^{3} + \frac{46986306875250069037307}{95407194748627202221008} a^{2} - \frac{8577380107520144328385}{47703597374313601110504} a + \frac{22247435218332142648681}{63604796499084801480672}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2896407.5795219596 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T9):

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

\(\Q(\sqrt{33}) \), 3.1.44.1, 3.1.588.1, 6.2.574992.1, 6.2.345138948.1 x2, 6.2.1380555792.2, 9.1.270588935232.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{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$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$