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Properties

Label 18.6.54498357372...0089.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 13^{14}$
Root discriminant $26.90$
Ramified primes $7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_4 \times C_3$ (as 18T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3991, 18499, -37440, 49166, -47255, 33475, -18390, 7647, -1994, -345, 930, -765, 498, -229, 101, -36, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 6*x^16 - 36*x^15 + 101*x^14 - 229*x^13 + 498*x^12 - 765*x^11 + 930*x^10 - 345*x^9 - 1994*x^8 + 7647*x^7 - 18390*x^6 + 33475*x^5 - 47255*x^4 + 49166*x^3 - 37440*x^2 + 18499*x - 3991)
 
gp: K = bnfinit(x^18 - 3*x^17 + 6*x^16 - 36*x^15 + 101*x^14 - 229*x^13 + 498*x^12 - 765*x^11 + 930*x^10 - 345*x^9 - 1994*x^8 + 7647*x^7 - 18390*x^6 + 33475*x^5 - 47255*x^4 + 49166*x^3 - 37440*x^2 + 18499*x - 3991, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 6 x^{16} - 36 x^{15} + 101 x^{14} - 229 x^{13} + 498 x^{12} - 765 x^{11} + 930 x^{10} - 345 x^{9} - 1994 x^{8} + 7647 x^{7} - 18390 x^{6} + 33475 x^{5} - 47255 x^{4} + 49166 x^{3} - 37440 x^{2} + 18499 x - 3991 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(54498357372899208270500089=7^{12}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.90$
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:  $7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{12} a^{12} - \frac{1}{6} a^{11} - \frac{1}{4} a^{10} + \frac{1}{6} a^{9} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{1}{12} a^{6} + \frac{1}{4} a^{4} - \frac{1}{3} a^{3} - \frac{5}{12} a^{2} - \frac{1}{6} a - \frac{5}{12}$, $\frac{1}{60} a^{13} - \frac{1}{60} a^{12} - \frac{1}{12} a^{11} + \frac{11}{60} a^{10} + \frac{13}{60} a^{9} - \frac{1}{20} a^{8} + \frac{11}{60} a^{7} + \frac{1}{60} a^{6} + \frac{1}{4} a^{5} + \frac{11}{60} a^{4} + \frac{1}{20} a^{3} + \frac{29}{60} a^{2} - \frac{19}{60} a - \frac{17}{60}$, $\frac{1}{60} a^{14} - \frac{1}{60} a^{12} - \frac{1}{15} a^{11} + \frac{3}{20} a^{10} - \frac{1}{6} a^{9} + \frac{1}{20} a^{8} + \frac{1}{30} a^{7} - \frac{3}{20} a^{6} - \frac{1}{15} a^{5} - \frac{1}{60} a^{4} - \frac{3}{10} a^{3} - \frac{1}{4} a^{2} - \frac{4}{15} a - \frac{1}{5}$, $\frac{1}{60} a^{15} - \frac{1}{10} a^{11} - \frac{7}{30} a^{10} - \frac{1}{15} a^{9} - \frac{1}{10} a^{8} - \frac{2}{15} a^{7} + \frac{1}{30} a^{6} + \frac{7}{30} a^{5} + \frac{2}{15} a^{4} + \frac{7}{15} a^{3} + \frac{3}{10} a^{2} + \frac{19}{60} a + \frac{3}{10}$, $\frac{1}{66540} a^{16} - \frac{67}{13308} a^{15} + \frac{33}{4436} a^{14} - \frac{5}{1109} a^{13} + \frac{863}{22180} a^{12} - \frac{1106}{16635} a^{11} - \frac{1703}{22180} a^{10} + \frac{2111}{16635} a^{9} - \frac{1763}{66540} a^{8} - \frac{69}{5545} a^{7} - \frac{7301}{66540} a^{6} + \frac{2999}{33270} a^{5} + \frac{6361}{22180} a^{4} + \frac{3982}{16635} a^{3} - \frac{11593}{33270} a^{2} + \frac{21913}{66540} a + \frac{53}{1109}$, $\frac{1}{27322115394298367511300} a^{17} - \frac{129732986973149077}{27322115394298367511300} a^{16} - \frac{763364174240855891}{4553685899049727918550} a^{15} - \frac{2378929975588267521}{2276842949524863959275} a^{14} + \frac{50792463629659660573}{9107371798099455837100} a^{13} - \frac{186722928074875964573}{5464423078859673502260} a^{12} - \frac{5013632658847732711507}{27322115394298367511300} a^{11} + \frac{340581106841455422193}{27322115394298367511300} a^{10} - \frac{703744088232988181849}{9107371798099455837100} a^{9} - \frac{1268605176793983001459}{9107371798099455837100} a^{8} + \frac{4230983467545533748499}{27322115394298367511300} a^{7} + \frac{2593204979361673092391}{27322115394298367511300} a^{6} + \frac{6821080853792939991551}{27322115394298367511300} a^{5} + \frac{10008239009375368361141}{27322115394298367511300} a^{4} - \frac{742752420413895464827}{2276842949524863959275} a^{3} - \frac{1950480001097108827303}{4553685899049727918550} a^{2} + \frac{3484842590158632634237}{27322115394298367511300} a + \frac{550895166619022887531}{27322115394298367511300}$

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:  $11$
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:  \( 1529227.92476 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times A_4$ (as 18T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 12 conjugacy class representatives for $A_4 \times C_3$
Character table for $A_4 \times C_3$

Intermediate fields

3.3.8281.2, 3.3.8281.1, 3.3.169.1, \(\Q(\zeta_{7})^+\), 6.2.405769.1, 9.9.567869252041.1

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 12 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$13$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$