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Properties

Label 18.18.4556898638...2256.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 3^{31}\cdot 23^{9}$
Root discriminant $50.50$
Ramified primes $2, 3, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times S_4$ (as 18T30)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, -420, -3390, 1902, 28668, 20367, -59961, -83943, 6090, 58964, 19524, -13659, -7833, 876, 1062, 39, -54, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 54*x^16 + 39*x^15 + 1062*x^14 + 876*x^13 - 7833*x^12 - 13659*x^11 + 19524*x^10 + 58964*x^9 + 6090*x^8 - 83943*x^7 - 59961*x^6 + 20367*x^5 + 28668*x^4 + 1902*x^3 - 3390*x^2 - 420*x + 19)
 
gp: K = bnfinit(x^18 - 3*x^17 - 54*x^16 + 39*x^15 + 1062*x^14 + 876*x^13 - 7833*x^12 - 13659*x^11 + 19524*x^10 + 58964*x^9 + 6090*x^8 - 83943*x^7 - 59961*x^6 + 20367*x^5 + 28668*x^4 + 1902*x^3 - 3390*x^2 - 420*x + 19, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 54 x^{16} + 39 x^{15} + 1062 x^{14} + 876 x^{13} - 7833 x^{12} - 13659 x^{11} + 19524 x^{10} + 58964 x^{9} + 6090 x^{8} - 83943 x^{7} - 59961 x^{6} + 20367 x^{5} + 28668 x^{4} + 1902 x^{3} - 3390 x^{2} - 420 x + 19 \)

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:  \(4556898638363530990028531552256=2^{12}\cdot 3^{31}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.50$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{20} a^{15} + \frac{1}{20} a^{14} - \frac{1}{4} a^{13} + \frac{1}{10} a^{12} - \frac{1}{10} a^{11} + \frac{3}{20} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{10} a^{7} - \frac{9}{20} a^{6} - \frac{3}{10} a^{5} + \frac{1}{10} a^{4} + \frac{1}{4} a^{3} - \frac{3}{10} a^{2} - \frac{1}{4} a - \frac{2}{5}$, $\frac{1}{340} a^{16} - \frac{2}{85} a^{15} + \frac{1}{340} a^{14} - \frac{9}{170} a^{13} + \frac{9}{68} a^{12} + \frac{41}{340} a^{11} + \frac{83}{340} a^{10} + \frac{1}{68} a^{9} - \frac{33}{85} a^{8} + \frac{29}{340} a^{7} + \frac{11}{68} a^{6} - \frac{109}{340} a^{5} - \frac{43}{340} a^{4} - \frac{161}{340} a^{3} + \frac{16}{85} a^{2} + \frac{1}{20} a + \frac{137}{340}$, $\frac{1}{3704563123397184604340} a^{17} - \frac{4238129363696297801}{3704563123397184604340} a^{16} - \frac{55874114308304040217}{3704563123397184604340} a^{15} - \frac{124150715001668591333}{3704563123397184604340} a^{14} + \frac{430405280648983257}{3069232082350608620} a^{13} + \frac{178993610281677339901}{1852281561698592302170} a^{12} + \frac{212699736752003802601}{926140780849296151085} a^{11} + \frac{3215937519712362579}{21791547784689321202} a^{10} - \frac{1655481404605607919297}{3704563123397184604340} a^{9} + \frac{89402122789958751305}{740912624679436920868} a^{8} + \frac{26756099891982243429}{54478869461723303005} a^{7} + \frac{224176877643655185717}{1852281561698592302170} a^{6} - \frac{855127407562247819197}{1852281561698592302170} a^{5} + \frac{364045086307928752841}{926140780849296151085} a^{4} + \frac{529424529259578603217}{3704563123397184604340} a^{3} + \frac{1772151926501314679637}{3704563123397184604340} a^{2} - \frac{207008987324236900606}{926140780849296151085} a + \frac{123237656423299298649}{740912624679436920868}$

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:  \( 5162525491.39 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_4$ (as 18T30):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 15 conjugacy class representatives for $C_3\times S_4$
Character table for $C_3\times S_4$

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.621.1, 6.6.425747664.1, 9.9.174583151469.1

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

Sibling fields

Degree 12 sibling: data not computed
Degree 18 sibling: 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
3Data not computed
$23$23.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$
23.12.9.2$x^{12} - 46 x^{8} + 529 x^{4} - 194672$$4$$3$$9$$D_4 \times C_3$$[\ ]_{4}^{6}$