Properties

Label 18.0.25299902311...6912.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{31}$
Root discriminant $10.53$
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: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 36*x^16 - 81*x^15 + 105*x^14 - 63*x^13 - 21*x^12 + 72*x^11 - 63*x^10 + 27*x^9 - 6*x^6 - 9*x^5 + 18*x^4 - 9*x^2 + 3)
 
gp: K = bnfinit(x^18 - 9*x^17 + 36*x^16 - 81*x^15 + 105*x^14 - 63*x^13 - 21*x^12 + 72*x^11 - 63*x^10 + 27*x^9 - 6*x^6 - 9*x^5 + 18*x^4 - 9*x^2 + 3, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 0, -9, 0, 18, -9, -6, 0, 0, 27, -63, 72, -21, -63, 105, -81, 36, -9, 1]);
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 36 x^{16} - 81 x^{15} + 105 x^{14} - 63 x^{13} - 21 x^{12} + 72 x^{11} - 63 x^{10} + 27 x^{9} - 6 x^{6} - 9 x^{5} + 18 x^{4} - 9 x^{2} + 3 \)

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

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-2529990231179046912=-\,2^{12}\cdot 3^{31}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $10.53$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $18$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{159031} a^{17} + \frac{66377}{159031} a^{16} + \frac{72610}{159031} a^{15} + \frac{57769}{159031} a^{14} + \frac{20374}{159031} a^{13} - \frac{10354}{159031} a^{12} - \frac{28683}{159031} a^{11} - \frac{71403}{159031} a^{10} + \frac{77396}{159031} a^{9} + \frac{37335}{159031} a^{8} + \frac{23175}{159031} a^{7} + \frac{29656}{159031} a^{6} - \frac{60570}{159031} a^{5} - \frac{60225}{159031} a^{4} - \frac{57492}{159031} a^{3} - \frac{78943}{159031} a^{2} - \frac{2433}{159031} a + \frac{58358}{159031}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{160828}{159031} a^{17} + \frac{1425060}{159031} a^{16} - \frac{5322773}{159031} a^{15} + \frac{10373365}{159031} a^{14} - \frac{9576808}{159031} a^{13} - \frac{636613}{159031} a^{12} + \frac{10672384}{159031} a^{11} - \frac{10840934}{159031} a^{10} + \frac{5478567}{159031} a^{9} - \frac{457006}{159031} a^{8} - \frac{1569663}{159031} a^{7} - \frac{652571}{159031} a^{6} + \frac{862241}{159031} a^{5} + \frac{3263865}{159031} a^{4} - \frac{1806367}{159031} a^{3} - \frac{1744422}{159031} a^{2} + \frac{555357}{159031} a + \frac{886289}{159031} \) (order $18$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 404.056392969 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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, \(\Q(\zeta_{9})^+\), 6.0.34992.1, 6.0.314928.1 x2, \(\Q(\zeta_{9})\), 9.3.918330048.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.314928.1
Degree 9 sibling: 9.3.918330048.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
3Data not computed