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Properties

Label 18.0.53072791277...0608.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 37^{12}$
Root discriminant $30.53$
Ramified primes $2, 3, 37$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 27, -8, 28, 161, -34, 76, 656, -653, 915, -201, 127, 97, 7, 10, 10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 10*x^16 + 10*x^15 + 7*x^14 + 97*x^13 + 127*x^12 - 201*x^11 + 915*x^10 - 653*x^9 + 656*x^8 + 76*x^7 - 34*x^6 + 161*x^5 + 28*x^4 - 8*x^3 + 27*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^18 - 2*x^17 + 10*x^16 + 10*x^15 + 7*x^14 + 97*x^13 + 127*x^12 - 201*x^11 + 915*x^10 - 653*x^9 + 656*x^8 + 76*x^7 - 34*x^6 + 161*x^5 + 28*x^4 - 8*x^3 + 27*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 10 x^{16} + 10 x^{15} + 7 x^{14} + 97 x^{13} + 127 x^{12} - 201 x^{11} + 915 x^{10} - 653 x^{9} + 656 x^{8} + 76 x^{7} - 34 x^{6} + 161 x^{5} + 28 x^{4} - 8 x^{3} + 27 x^{2} - 5 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-530727912779568801529540608=-\,2^{12}\cdot 3^{9}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.53$
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:  $2, 3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{73} a^{15} - \frac{20}{73} a^{14} - \frac{23}{73} a^{13} - \frac{17}{73} a^{12} + \frac{26}{73} a^{11} - \frac{13}{73} a^{10} + \frac{6}{73} a^{9} + \frac{17}{73} a^{8} + \frac{22}{73} a^{7} - \frac{12}{73} a^{6} + \frac{29}{73} a^{5} - \frac{13}{73} a^{4} + \frac{12}{73} a^{3} - \frac{31}{73} a^{2} - \frac{23}{73} a - \frac{7}{73}$, $\frac{1}{26207} a^{16} + \frac{64}{26207} a^{15} + \frac{2750}{26207} a^{14} + \frac{3964}{26207} a^{13} + \frac{7212}{26207} a^{12} + \frac{4361}{26207} a^{11} + \frac{4243}{26207} a^{10} - \frac{4881}{26207} a^{9} + \frac{11451}{26207} a^{8} + \frac{7895}{26207} a^{7} - \frac{2074}{26207} a^{6} + \frac{10526}{26207} a^{5} - \frac{6336}{26207} a^{4} - \frac{9608}{26207} a^{3} - \frac{11898}{26207} a^{2} + \frac{7624}{26207} a - \frac{10224}{26207}$, $\frac{1}{77342493889837} a^{17} + \frac{173902510}{77342493889837} a^{16} + \frac{368441889202}{77342493889837} a^{15} - \frac{23991969407335}{77342493889837} a^{14} + \frac{2196453573100}{7031135808167} a^{13} - \frac{7808024262656}{77342493889837} a^{12} - \frac{3292906144494}{7031135808167} a^{11} - \frac{26660910076408}{77342493889837} a^{10} - \frac{28103565643721}{77342493889837} a^{9} + \frac{9061480485285}{77342493889837} a^{8} - \frac{267222947268}{77342493889837} a^{7} + \frac{10695437565553}{77342493889837} a^{6} + \frac{216841324036}{7031135808167} a^{5} - \frac{1229390572326}{77342493889837} a^{4} - \frac{8896574119120}{77342493889837} a^{3} - \frac{21391925429769}{77342493889837} a^{2} - \frac{9815363835570}{77342493889837} a + \frac{13686466629766}{77342493889837}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{61546276048}{215438701643} a^{17} - \frac{130223656922}{215438701643} a^{16} + \frac{618459364454}{215438701643} a^{15} + \frac{568332425901}{215438701643} a^{14} + \frac{22495432094}{19585336513} a^{13} + \frac{5816916261874}{215438701643} a^{12} + \frac{645308973352}{19585336513} a^{11} - \frac{14308658288860}{215438701643} a^{10} + \frac{773988804288}{2951215091} a^{9} - \frac{43873847916768}{215438701643} a^{8} + \frac{34954800613098}{215438701643} a^{7} + \frac{8065666913716}{215438701643} a^{6} - \frac{716989355282}{19585336513} a^{5} + \frac{8816699348122}{215438701643} a^{4} + \frac{2110444676338}{215438701643} a^{3} - \frac{1284894236004}{215438701643} a^{2} + \frac{1378518139828}{215438701643} a - \frac{33377666261}{215438701643} \) (order $6$)
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:  \( 174504.17013875535 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.1369.1, 3.1.5476.1, 6.0.50602347.1, 6.0.809637552.2, 9.3.164206490176.1

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 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$37$37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$