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Properties

Label 18.0.93265559882...3968.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{33}$
Root discriminant $18.88$
Ramified primes $2, 3$
Class number $3$
Class group $[3]$
Galois group $C_2\times He_3:C_2$ (as 18T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, 3, 18, 39, 21, 3, -42, 48, -2, -39, 15, -9, 6, -3, -3, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 6*x^16 - 3*x^15 - 3*x^14 + 6*x^13 - 9*x^12 + 15*x^11 - 39*x^10 - 2*x^9 + 48*x^8 - 42*x^7 + 3*x^6 + 21*x^5 + 39*x^4 + 18*x^3 + 3*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 + 6*x^16 - 3*x^15 - 3*x^14 + 6*x^13 - 9*x^12 + 15*x^11 - 39*x^10 - 2*x^9 + 48*x^8 - 42*x^7 + 3*x^6 + 21*x^5 + 39*x^4 + 18*x^3 + 3*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 6 x^{16} - 3 x^{15} - 3 x^{14} + 6 x^{13} - 9 x^{12} + 15 x^{11} - 39 x^{10} - 2 x^{9} + 48 x^{8} - 42 x^{7} + 3 x^{6} + 21 x^{5} + 39 x^{4} + 18 x^{3} + 3 x^{2} + 3 x + 1 \)

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:  \(-93265559882184385363968=-\,2^{24}\cdot 3^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.88$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{9} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{9}$, $\frac{1}{9} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{9} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{12} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{13} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{9} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{14} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{9} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{15} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{9} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{54} a^{16} + \frac{1}{27} a^{15} - \frac{1}{18} a^{14} - \frac{1}{27} a^{13} + \frac{1}{27} a^{12} + \frac{1}{54} a^{10} + \frac{1}{27} a^{9} - \frac{8}{27} a^{7} + \frac{2}{27} a^{6} - \frac{4}{9} a^{5} + \frac{5}{54} a^{4} - \frac{7}{27} a^{3} + \frac{1}{3} a^{2} + \frac{10}{27} a + \frac{13}{54}$, $\frac{1}{1242} a^{17} + \frac{1}{414} a^{16} + \frac{47}{1242} a^{15} + \frac{49}{1242} a^{14} - \frac{1}{23} a^{13} + \frac{25}{621} a^{12} + \frac{61}{1242} a^{11} - \frac{11}{414} a^{10} + \frac{31}{621} a^{9} + \frac{208}{621} a^{8} + \frac{79}{207} a^{7} - \frac{232}{621} a^{6} + \frac{71}{1242} a^{5} + \frac{65}{138} a^{4} - \frac{238}{621} a^{3} + \frac{7}{621} a^{2} + \frac{167}{414} a - \frac{533}{1242}$

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:  \( -\frac{33}{23} a^{17} + \frac{995}{207} a^{16} - \frac{2137}{207} a^{15} + \frac{554}{69} a^{14} + \frac{260}{207} a^{13} - \frac{1832}{207} a^{12} + \frac{3319}{207} a^{11} - \frac{1885}{69} a^{10} + \frac{13648}{207} a^{9} - \frac{480}{23} a^{8} - \frac{12392}{207} a^{7} + \frac{16690}{207} a^{6} - \frac{779}{23} a^{5} - \frac{3407}{207} a^{4} - \frac{10612}{207} a^{3} - \frac{1720}{207} a^{2} - \frac{241}{69} a - \frac{652}{207} \) (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:  \( 28130.578224 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.648.1, 6.0.1259712.1, 9.1.176319369216.3

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
3Data not computed