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Properties

Label 18.0.35371827155...6416.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{34}\cdot 3^{30}$
Root discriminant $23.11$
Ramified primes $2, 3$
Class number $3$
Class group $[3]$
Galois group $C_2\times S_3^2$ (as 18T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18, 0, 0, 18, 0, -54, 9, 18, -36, 6, 99, -18, 0, 54, -9, -18, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 18*x^16 - 18*x^15 - 9*x^14 + 54*x^13 - 18*x^11 + 99*x^10 + 6*x^9 - 36*x^8 + 18*x^7 + 9*x^6 - 54*x^5 + 18*x^3 + 18)
 
gp: K = bnfinit(x^18 - 6*x^17 + 18*x^16 - 18*x^15 - 9*x^14 + 54*x^13 - 18*x^11 + 99*x^10 + 6*x^9 - 36*x^8 + 18*x^7 + 9*x^6 - 54*x^5 + 18*x^3 + 18, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 18 x^{16} - 18 x^{15} - 9 x^{14} + 54 x^{13} - 18 x^{11} + 99 x^{10} + 6 x^{9} - 36 x^{8} + 18 x^{7} + 9 x^{6} - 54 x^{5} + 18 x^{3} + 18 \)

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:  \(-3537182715531733726396416=-\,2^{34}\cdot 3^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.11$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{30} a^{12} - \frac{2}{15} a^{11} + \frac{1}{10} a^{10} - \frac{2}{15} a^{9} - \frac{3}{10} a^{8} - \frac{1}{2} a^{6} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{30} a^{13} - \frac{1}{10} a^{11} - \frac{1}{15} a^{10} - \frac{1}{6} a^{9} - \frac{1}{5} a^{8} - \frac{1}{2} a^{7} - \frac{1}{5} a^{5} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{30} a^{14} - \frac{2}{15} a^{11} + \frac{2}{15} a^{10} + \frac{1}{15} a^{9} - \frac{2}{5} a^{8} + \frac{3}{10} a^{6} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{30} a^{15} - \frac{1}{15} a^{11} + \frac{2}{15} a^{10} + \frac{1}{15} a^{9} - \frac{1}{5} a^{8} + \frac{3}{10} a^{7} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{30} a^{16} - \frac{2}{15} a^{11} - \frac{1}{15} a^{10} - \frac{2}{15} a^{9} - \frac{3}{10} a^{8} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{114540} a^{17} + \frac{613}{114540} a^{16} + \frac{1483}{114540} a^{15} + \frac{1639}{114540} a^{14} - \frac{88}{9545} a^{13} - \frac{61}{9545} a^{12} - \frac{1488}{9545} a^{11} - \frac{3761}{28635} a^{10} + \frac{3811}{22908} a^{9} - \frac{1953}{7636} a^{8} + \frac{14619}{38180} a^{7} - \frac{18583}{38180} a^{6} - \frac{1529}{19090} a^{5} + \frac{2111}{9545} a^{4} + \frac{953}{9545} a^{3} - \frac{7581}{19090} a^{2} + \frac{7319}{19090} a - \frac{5323}{19090}$

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:  \( -\frac{22}{28635} a^{17} + \frac{1786}{28635} a^{16} - \frac{9718}{28635} a^{15} + \frac{17323}{19090} a^{14} - \frac{5619}{9545} a^{13} - \frac{27803}{28635} a^{12} + \frac{22131}{9545} a^{11} + \frac{21236}{9545} a^{10} - \frac{39319}{28635} a^{9} + \frac{20112}{9545} a^{8} + \frac{23911}{9545} a^{7} - \frac{31853}{19090} a^{6} - \frac{43446}{9545} a^{5} + \frac{7041}{9545} a^{4} - \frac{3686}{9545} a^{3} - \frac{10056}{9545} a^{2} + \frac{6312}{9545} a + \frac{16586}{9545} \) (order $4$)
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:  \( 233467.89303899015 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3^2$ (as 18T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 18 conjugacy class representatives for $C_2\times S_3^2$
Character table for $C_2\times S_3^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.108.1, 3.1.648.1, 6.0.186624.1, 6.0.6718464.3, 9.1.117546246144.4

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

Sibling fields

Degree 12 siblings: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.26.64$x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{6} - 2 x^{4} + 4 x^{3} + 2$$12$$1$$26$$S_3 \times C_2^2$$[2, 3]_{3}^{2}$
3Data not computed