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Properties

Label 18.0.59303834790...3088.5
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 7^{12}$
Root discriminant $20.93$
Ramified primes $2, 3, 7$
Class number $3$
Class group $[3]$
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![169, -1209, 6153, -11200, 9858, -5658, 4514, -5184, 4887, -3423, 1791, -792, 498, -426, 294, -140, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 140*x^15 + 294*x^14 - 426*x^13 + 498*x^12 - 792*x^11 + 1791*x^10 - 3423*x^9 + 4887*x^8 - 5184*x^7 + 4514*x^6 - 5658*x^5 + 9858*x^4 - 11200*x^3 + 6153*x^2 - 1209*x + 169)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 140*x^15 + 294*x^14 - 426*x^13 + 498*x^12 - 792*x^11 + 1791*x^10 - 3423*x^9 + 4887*x^8 - 5184*x^7 + 4514*x^6 - 5658*x^5 + 9858*x^4 - 11200*x^3 + 6153*x^2 - 1209*x + 169, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 140 x^{15} + 294 x^{14} - 426 x^{13} + 498 x^{12} - 792 x^{11} + 1791 x^{10} - 3423 x^{9} + 4887 x^{8} - 5184 x^{7} + 4514 x^{6} - 5658 x^{5} + 9858 x^{4} - 11200 x^{3} + 6153 x^{2} - 1209 x + 169 \)

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:  \(-593038347905528851673088=-\,2^{12}\cdot 3^{21}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.93$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{6} + \frac{1}{6} a^{3} + \frac{1}{6}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{7} + \frac{1}{6} a^{4} + \frac{1}{6} a$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{8} + \frac{1}{6} a^{5} + \frac{1}{6} a^{2}$, $\frac{1}{156} a^{12} + \frac{5}{78} a^{11} + \frac{3}{52} a^{10} + \frac{1}{13} a^{9} - \frac{5}{78} a^{8} + \frac{5}{26} a^{7} + \frac{9}{52} a^{6} + \frac{5}{78} a^{5} - \frac{5}{26} a^{4} - \frac{10}{39} a^{3} - \frac{23}{156} a^{2} + \frac{5}{26} a - \frac{5}{12}$, $\frac{1}{156} a^{13} - \frac{1}{12} a^{11} - \frac{1}{6} a^{8} - \frac{1}{4} a^{7} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{4} a^{3} + \frac{1}{6} a^{2} - \frac{53}{156} a$, $\frac{1}{156} a^{14} - \frac{1}{12} a^{10} - \frac{1}{4} a^{8} - \frac{1}{6} a^{7} - \frac{1}{4} a^{6} - \frac{1}{3} a^{5} - \frac{1}{12} a^{4} + \frac{16}{39} a^{2} + \frac{1}{6} a - \frac{1}{4}$, $\frac{1}{468} a^{15} + \frac{1}{468} a^{14} - \frac{1}{468} a^{12} - \frac{23}{468} a^{11} - \frac{11}{234} a^{10} - \frac{25}{468} a^{9} + \frac{23}{468} a^{8} - \frac{17}{468} a^{7} + \frac{5}{26} a^{6} + \frac{9}{52} a^{5} - \frac{139}{468} a^{4} + \frac{5}{18} a^{3} - \frac{121}{468} a^{2} + \frac{35}{468} a + \frac{5}{18}$, $\frac{1}{468} a^{16} - \frac{1}{468} a^{14} - \frac{1}{468} a^{13} - \frac{1}{468} a^{12} - \frac{23}{468} a^{11} + \frac{5}{78} a^{10} - \frac{1}{39} a^{9} - \frac{4}{117} a^{8} - \frac{43}{468} a^{7} - \frac{11}{78} a^{6} + \frac{56}{117} a^{5} - \frac{49}{468} a^{4} + \frac{1}{468} a^{3} - \frac{31}{156} a^{2} + \frac{101}{468} a + \frac{5}{36}$, $\frac{1}{503657647404444} a^{17} - \frac{51864876737}{83942941234074} a^{16} - \frac{105125338103}{251828823702222} a^{15} - \frac{156440431519}{125914411851111} a^{14} + \frac{293203809767}{503657647404444} a^{13} + \frac{1568872944673}{503657647404444} a^{12} + \frac{4041827535765}{55961960822716} a^{11} + \frac{2759241259497}{55961960822716} a^{10} - \frac{7303808137309}{503657647404444} a^{9} + \frac{4237690389965}{125914411851111} a^{8} + \frac{11674571667527}{167885882468148} a^{7} - \frac{65935143995869}{503657647404444} a^{6} - \frac{4910566921159}{125914411851111} a^{5} - \frac{14142041403967}{251828823702222} a^{4} - \frac{6604169349989}{167885882468148} a^{3} - \frac{94718679509359}{503657647404444} a^{2} + \frac{90970548699541}{251828823702222} a + \frac{3963939922063}{12914298651396}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( \frac{46125936002}{41971470617037} a^{17} - \frac{274930706137}{41971470617037} a^{16} + \frac{960672549560}{41971470617037} a^{15} - \frac{1395471863480}{41971470617037} a^{14} - \frac{229311004900}{41971470617037} a^{13} + \frac{4897242058892}{41971470617037} a^{12} - \frac{5863926921544}{41971470617037} a^{11} - \frac{4731915303496}{41971470617037} a^{10} + \frac{12033402039710}{41971470617037} a^{9} + \frac{10056569093210}{41971470617037} a^{8} - \frac{50488038827560}{41971470617037} a^{7} + \frac{97088338790288}{41971470617037} a^{6} - \frac{27320354129460}{13990490205679} a^{5} - \frac{20101946712544}{41971470617037} a^{4} - \frac{33009084563608}{41971470617037} a^{3} + \frac{99087917632720}{13990490205679} a^{2} - \frac{262950778875830}{41971470617037} a + \frac{4210839445855}{3228574662849} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 51834.4889774 \)
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

\(\Q(\sqrt{-3}) \), 3.1.5292.1 x3, \(\Q(\zeta_{7})^+\), 6.0.84015792.1, 6.0.1714608.2 x2, 6.0.64827.1, 9.3.148203857088.5 x3

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

Sibling fields

Degree 6 sibling: 6.0.1714608.2
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
3Data not computed
$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}$