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Properties

Label 18.0.16599265906...9632.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{39}$
Root discriminant $17.16$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37, -36, -99, 87, 279, -459, 69, 261, 135, -944, 1458, -1449, 1119, -711, 369, -150, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 150*x^15 + 369*x^14 - 711*x^13 + 1119*x^12 - 1449*x^11 + 1458*x^10 - 944*x^9 + 135*x^8 + 261*x^7 + 69*x^6 - 459*x^5 + 279*x^4 + 87*x^3 - 99*x^2 - 36*x + 37)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 150*x^15 + 369*x^14 - 711*x^13 + 1119*x^12 - 1449*x^11 + 1458*x^10 - 944*x^9 + 135*x^8 + 261*x^7 + 69*x^6 - 459*x^5 + 279*x^4 + 87*x^3 - 99*x^2 - 36*x + 37, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 150 x^{15} + 369 x^{14} - 711 x^{13} + 1119 x^{12} - 1449 x^{11} + 1458 x^{10} - 944 x^{9} + 135 x^{8} + 261 x^{7} + 69 x^{6} - 459 x^{5} + 279 x^{4} + 87 x^{3} - 99 x^{2} - 36 x + 37 \)

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:  \(-16599265906765726789632=-\,2^{12}\cdot 3^{39}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.16$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{33337} a^{16} - \frac{18}{33337} a^{15} + \frac{1157}{33337} a^{14} + \frac{5674}{33337} a^{13} + \frac{15006}{33337} a^{12} - \frac{12711}{33337} a^{11} + \frac{2971}{33337} a^{10} - \frac{2307}{33337} a^{9} + \frac{298}{901} a^{8} + \frac{8425}{33337} a^{7} - \frac{2134}{33337} a^{6} - \frac{11000}{33337} a^{5} + \frac{5315}{33337} a^{4} + \frac{2861}{33337} a^{3} - \frac{10107}{33337} a^{2} + \frac{164}{629} a - \frac{331}{901}$, $\frac{1}{10958722093511} a^{17} - \frac{84256642}{10958722093511} a^{16} - \frac{4959136058055}{10958722093511} a^{15} - \frac{2533206974919}{10958722093511} a^{14} + \frac{634065718798}{10958722093511} a^{13} - \frac{5392394680219}{10958722093511} a^{12} - \frac{2291008894789}{10958722093511} a^{11} + \frac{370860990890}{10958722093511} a^{10} + \frac{4406881466785}{10958722093511} a^{9} - \frac{5284779676043}{10958722093511} a^{8} - \frac{1400866361447}{10958722093511} a^{7} + \frac{3136213660618}{10958722093511} a^{6} + \frac{1268654233296}{10958722093511} a^{5} + \frac{3455544602950}{10958722093511} a^{4} + \frac{5012277225302}{10958722093511} a^{3} - \frac{3556536332192}{10958722093511} a^{2} - \frac{214515331562}{10958722093511} a + \frac{1624090626}{17422451659}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{838101743968}{10958722093511} a^{17} + \frac{6806609602755}{10958722093511} a^{16} - \frac{31325313759506}{10958722093511} a^{15} + \frac{95076188717768}{10958722093511} a^{14} - \frac{211686994831810}{10958722093511} a^{13} + \frac{368233076132381}{10958722093511} a^{12} - \frac{521563154055952}{10958722093511} a^{11} + \frac{592768317563272}{10958722093511} a^{10} - \frac{27179126381740}{644630711383} a^{9} + \frac{101345106160873}{10958722093511} a^{8} + \frac{221277975874344}{10958722093511} a^{7} - \frac{7804100427328}{644630711383} a^{6} - \frac{198102158301674}{10958722093511} a^{5} + \frac{14067328321184}{644630711383} a^{4} + \frac{8995680569872}{10958722093511} a^{3} - \frac{93084668224022}{10958722093511} a^{2} - \frac{900186154544}{644630711383} a + \frac{864693259789}{296181678203} \) (order $18$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 47261.4483684 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.972.2 x3, \(\Q(\zeta_{9})^+\), 6.0.2834352.2, 6.0.2834352.4 x2, \(\Q(\zeta_{9})\), 9.3.74384733888.4 x3

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

Sibling fields

Degree 6 sibling: 6.0.2834352.4
Degree 9 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
3Data not computed