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Properties

Label 18.0.59303834790...3088.3
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![1, -6, -9, 82, 60, -441, -106, 1170, -168, -1545, 651, 858, -435, -273, 135, 47, -18, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 18*x^16 + 47*x^15 + 135*x^14 - 273*x^13 - 435*x^12 + 858*x^11 + 651*x^10 - 1545*x^9 - 168*x^8 + 1170*x^7 - 106*x^6 - 441*x^5 + 60*x^4 + 82*x^3 - 9*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 - 18*x^16 + 47*x^15 + 135*x^14 - 273*x^13 - 435*x^12 + 858*x^11 + 651*x^10 - 1545*x^9 - 168*x^8 + 1170*x^7 - 106*x^6 - 441*x^5 + 60*x^4 + 82*x^3 - 9*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 18 x^{16} + 47 x^{15} + 135 x^{14} - 273 x^{13} - 435 x^{12} + 858 x^{11} + 651 x^{10} - 1545 x^{9} - 168 x^{8} + 1170 x^{7} - 106 x^{6} - 441 x^{5} + 60 x^{4} + 82 x^{3} - 9 x^{2} - 6 x + 1 \)

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:  \(-593038347905528851673088=-\,2^{12}\cdot 3^{21}\cdot 7^{12}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{96559} a^{16} - \frac{14237}{96559} a^{15} - \frac{46094}{96559} a^{14} + \frac{26141}{96559} a^{13} + \frac{20754}{96559} a^{12} + \frac{27693}{96559} a^{11} + \frac{35968}{96559} a^{10} + \frac{14277}{96559} a^{9} - \frac{43512}{96559} a^{8} + \frac{21086}{96559} a^{7} - \frac{9786}{96559} a^{6} - \frac{28234}{96559} a^{5} - \frac{14006}{96559} a^{4} + \frac{30469}{96559} a^{3} + \frac{39299}{96559} a^{2} + \frac{7105}{96559} a + \frac{17382}{96559}$, $\frac{1}{18015688543} a^{17} + \frac{70406}{18015688543} a^{16} + \frac{5883384264}{18015688543} a^{15} - \frac{7912762279}{18015688543} a^{14} - \frac{6143252766}{18015688543} a^{13} - \frac{3739526324}{18015688543} a^{12} + \frac{1934837525}{18015688543} a^{11} + \frac{7390477732}{18015688543} a^{10} - \frac{7590534276}{18015688543} a^{9} - \frac{7138811740}{18015688543} a^{8} + \frac{2777881827}{18015688543} a^{7} - \frac{8272077942}{18015688543} a^{6} - \frac{1131853816}{18015688543} a^{5} + \frac{41109588}{18015688543} a^{4} - \frac{6657324279}{18015688543} a^{3} + \frac{3490832339}{18015688543} a^{2} + \frac{3769699805}{18015688543} a + \frac{353207965}{18015688543}$

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:  \( -\frac{54060390}{41606671} a^{17} + \frac{143683529}{41606671} a^{16} + \frac{1017355500}{41606671} a^{15} - \frac{2183233208}{41606671} a^{14} - \frac{7945058386}{41606671} a^{13} + \frac{11909319407}{41606671} a^{12} + \frac{26760496582}{41606671} a^{11} - \frac{36669377142}{41606671} a^{10} - \frac{44813447562}{41606671} a^{9} + \frac{66516087241}{41606671} a^{8} + \frac{26380576658}{41606671} a^{7} - \frac{50801859282}{41606671} a^{6} - \frac{6948261628}{41606671} a^{5} + \frac{18305872801}{41606671} a^{4} + \frac{1318912502}{41606671} a^{3} - \frac{2971375452}{41606671} a^{2} - \frac{262374442}{41606671} a + \frac{178192837}{41606671} \) (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:  \( 20971.2187411 \)
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

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, \(\Q(\zeta_{7})^+\), 6.0.34992.1, 6.0.84015792.2 x2, 6.0.64827.1, 9.3.148203857088.2 x3

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

Sibling fields

Degree 6 sibling: data not computed
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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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}$