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Properties

Label 18.18.5431100773...8709.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{9}\cdot 7^{15}\cdot 491^{4}$
Root discriminant $34.74$
Ramified primes $3, 7, 491$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^3:C_6$ (as 18T85)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1889, -5735, 8245, 30557, -15218, -60382, 18292, 58009, -15041, -29680, 7445, 8512, -2127, -1372, 343, 116, -29, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 29*x^16 + 116*x^15 + 343*x^14 - 1372*x^13 - 2127*x^12 + 8512*x^11 + 7445*x^10 - 29680*x^9 - 15041*x^8 + 58009*x^7 + 18292*x^6 - 60382*x^5 - 15218*x^4 + 30557*x^3 + 8245*x^2 - 5735*x - 1889)
 
gp: K = bnfinit(x^18 - 4*x^17 - 29*x^16 + 116*x^15 + 343*x^14 - 1372*x^13 - 2127*x^12 + 8512*x^11 + 7445*x^10 - 29680*x^9 - 15041*x^8 + 58009*x^7 + 18292*x^6 - 60382*x^5 - 15218*x^4 + 30557*x^3 + 8245*x^2 - 5735*x - 1889, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 29 x^{16} + 116 x^{15} + 343 x^{14} - 1372 x^{13} - 2127 x^{12} + 8512 x^{11} + 7445 x^{10} - 29680 x^{9} - 15041 x^{8} + 58009 x^{7} + 18292 x^{6} - 60382 x^{5} - 15218 x^{4} + 30557 x^{3} + 8245 x^{2} - 5735 x - 1889 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5431100773839594625584038709=3^{9}\cdot 7^{15}\cdot 491^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.74$
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:  $3, 7, 491$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{2914680705628783864084} a^{17} + \frac{333781078350818601937}{2914680705628783864084} a^{16} + \frac{83402794456231611716}{728670176407195966021} a^{15} - \frac{2292668335258559385}{224206208125291066468} a^{14} - \frac{607925930267043179731}{1457340352814391932042} a^{13} - \frac{176036351492350861467}{1457340352814391932042} a^{12} - \frac{328643051688140992830}{728670176407195966021} a^{11} + \frac{37753217215705661757}{224206208125291066468} a^{10} - \frac{1183469147490514877477}{2914680705628783864084} a^{9} + \frac{47585200543723564671}{1457340352814391932042} a^{8} + \frac{167377607345458081161}{2914680705628783864084} a^{7} + \frac{511082817765557111603}{2914680705628783864084} a^{6} + \frac{334912464296559549557}{728670176407195966021} a^{5} - \frac{10810449288367277758}{56051552031322766617} a^{4} + \frac{578180270614225466465}{2914680705628783864084} a^{3} - \frac{13721890626214551537}{224206208125291066468} a^{2} + \frac{818128049429940111463}{2914680705628783864084} a + \frac{333917531045608589589}{728670176407195966021}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:  \( 131594864.531 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^3:C_6$ (as 18T85):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 162
The 13 conjugacy class representatives for $C_3^3:C_6$
Character table for $C_3^3:C_6$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{21})^+\), 9.9.5360595389541.1 x3

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

Sibling fields

Degree 9 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
491Data not computed