Properties

Label 18.0.40525551530...6267.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{39}$
Root discriminant $10.81$
Ramified prime $3$
Class number $1$
Class group Trivial
Galois Group $S_3 \times C_3$ (as 18T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 6, 0, 0, 33, 0, 0, 20, 0, 0, 15, 0, 0, -3, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^15 + 15*x^12 + 20*x^9 + 33*x^6 + 6*x^3 + 1)
gp: K = bnfinit(x^18 - 3*x^15 + 15*x^12 + 20*x^9 + 33*x^6 + 6*x^3 + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 3 x^{15} \) \(\mathstrut +\mathstrut 15 x^{12} \) \(\mathstrut +\mathstrut 20 x^{9} \) \(\mathstrut +\mathstrut 33 x^{6} \) \(\mathstrut +\mathstrut 6 x^{3} \) \(\mathstrut +\mathstrut 1 \)

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:  \(-4052555153018976267=-\,3^{39}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.81$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{2} - \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{9} + \frac{1}{9} a^{3} + \frac{1}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{10} + \frac{1}{9} a^{4} + \frac{1}{9} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{5} + \frac{1}{9} a - \frac{1}{9}$, $\frac{1}{171} a^{15} - \frac{5}{171} a^{12} + \frac{2}{57} a^{9} - \frac{11}{171} a^{6} - \frac{2}{171} a^{3} + \frac{16}{57}$, $\frac{1}{513} a^{16} + \frac{1}{513} a^{15} + \frac{14}{513} a^{13} + \frac{14}{513} a^{12} + \frac{25}{513} a^{10} + \frac{25}{513} a^{9} + \frac{46}{513} a^{7} + \frac{46}{513} a^{6} + \frac{131}{513} a^{4} + \frac{131}{513} a^{3} + \frac{124}{513} a + \frac{124}{513}$, $\frac{1}{513} a^{17} - \frac{1}{513} a^{15} + \frac{14}{513} a^{14} - \frac{14}{513} a^{12} + \frac{25}{513} a^{11} - \frac{25}{513} a^{9} + \frac{46}{513} a^{8} - \frac{46}{513} a^{6} + \frac{131}{513} a^{5} - \frac{131}{513} a^{3} + \frac{124}{513} a^{2} - \frac{124}{513}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( \frac{5}{171} a^{15} - \frac{2}{57} a^{12} + \frac{49}{171} a^{9} + \frac{230}{171} a^{6} + \frac{136}{57} a^{3} + \frac{202}{171} \) (order $18$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{31}{171} a^{16} - \frac{98}{171} a^{13} + \frac{157}{57} a^{10} + \frac{571}{171} a^{7} + \frac{793}{171} a^{4} - \frac{17}{57} a \),  \( \frac{47}{171} a^{16} - \frac{53}{57} a^{13} + \frac{757}{171} a^{10} + \frac{680}{171} a^{7} + \frac{131}{19} a^{4} - \frac{347}{171} a \),  \( \frac{67}{171} a^{17} + \frac{104}{513} a^{16} - \frac{49}{513} a^{15} - \frac{202}{171} a^{14} - \frac{368}{513} a^{13} + \frac{169}{513} a^{12} + \frac{1010}{171} a^{11} + \frac{1745}{513} a^{10} - \frac{826}{513} a^{9} + \frac{1315}{171} a^{8} + \frac{1193}{513} a^{7} - \frac{544}{513} a^{6} + \frac{2222}{171} a^{5} + \frac{2566}{513} a^{4} - \frac{1631}{513} a^{3} + \frac{404}{171} a^{2} - \frac{784}{513} a + \frac{137}{513} \),  \( \frac{5}{171} a^{17} - \frac{127}{513} a^{16} - \frac{40}{513} a^{15} - \frac{25}{171} a^{14} + \frac{388}{513} a^{13} + \frac{124}{513} a^{12} + \frac{106}{171} a^{11} - \frac{1921}{513} a^{10} - \frac{601}{513} a^{9} - \frac{55}{171} a^{8} - \frac{2422}{513} a^{7} - \frac{814}{513} a^{6} - \frac{10}{171} a^{5} - \frac{4040}{513} a^{4} - \frac{965}{513} a^{3} - \frac{368}{171} a^{2} - \frac{301}{513} a + \frac{56}{513} \),  \( \frac{55}{513} a^{17} + \frac{86}{513} a^{15} - \frac{199}{513} a^{14} - \frac{278}{513} a^{12} + \frac{919}{513} a^{11} + \frac{1352}{513} a^{9} + \frac{649}{513} a^{8} + \frac{1391}{513} a^{6} + \frac{935}{513} a^{5} + \frac{2602}{513} a^{3} - \frac{647}{513} a^{2} + \frac{119}{513} \),  \( \frac{37}{513} a^{17} - \frac{34}{513} a^{16} - \frac{71}{513} a^{15} - \frac{109}{513} a^{14} + \frac{94}{513} a^{13} + \frac{203}{513} a^{12} + \frac{526}{513} a^{11} - \frac{508}{513} a^{10} - \frac{1034}{513} a^{9} + \frac{847}{513} a^{8} - \frac{709}{513} a^{7} - \frac{1556}{513} a^{6} + \frac{971}{513} a^{5} - \frac{1661}{513} a^{4} - \frac{2632}{513} a^{3} - \frac{257}{513} a^{2} - \frac{454}{513} a - \frac{197}{513} \),  \( \frac{67}{171} a^{17} + \frac{40}{513} a^{16} + \frac{37}{513} a^{15} - \frac{202}{171} a^{14} - \frac{124}{513} a^{13} - \frac{109}{513} a^{12} + \frac{1010}{171} a^{11} + \frac{601}{513} a^{10} + \frac{526}{513} a^{9} + \frac{1315}{171} a^{8} + \frac{814}{513} a^{7} + \frac{847}{513} a^{6} + \frac{2222}{171} a^{5} + \frac{965}{513} a^{4} + \frac{971}{513} a^{3} + \frac{404}{171} a^{2} + \frac{457}{513} a - \frac{257}{513} \),  \( \frac{8}{19} a^{17} + \frac{89}{513} a^{16} - \frac{1}{27} a^{15} - \frac{227}{171} a^{14} - \frac{293}{513} a^{13} + \frac{4}{27} a^{12} + \frac{124}{19} a^{11} + \frac{1427}{513} a^{10} - \frac{19}{27} a^{9} + \frac{140}{19} a^{8} + \frac{1358}{513} a^{7} - \frac{1}{27} a^{6} + \frac{2212}{171} a^{5} + \frac{2596}{513} a^{4} - \frac{23}{27} a^{3} + \frac{4}{19} a^{2} - \frac{193}{513} a + \frac{8}{27} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 529.942929585 \)
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

\(\Q(\sqrt{-3}) \), 3.1.243.1 x3, \(\Q(\zeta_{9})^+\), 6.0.177147.2, 6.0.177147.1 x2, \(\Q(\zeta_{9})\), 9.3.1162261467.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.177147.1
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ 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.

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
3Data not computed