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Properties

Label 18.0.18783281539...0208.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 13^{12}$
Root discriminant $15.20$
Ramified primes $2, 3, 13$
Class number $1$
Class group Trivial
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, 22, -59, 110, -126, 64, 20, -28, 5, -28, 20, 64, -126, 110, -59, 22, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 22*x^16 - 59*x^15 + 110*x^14 - 126*x^13 + 64*x^12 + 20*x^11 - 28*x^10 + 5*x^9 - 28*x^8 + 20*x^7 + 64*x^6 - 126*x^5 + 110*x^4 - 59*x^3 + 22*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 22*x^16 - 59*x^15 + 110*x^14 - 126*x^13 + 64*x^12 + 20*x^11 - 28*x^10 + 5*x^9 - 28*x^8 + 20*x^7 + 64*x^6 - 126*x^5 + 110*x^4 - 59*x^3 + 22*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 22 x^{16} - 59 x^{15} + 110 x^{14} - 126 x^{13} + 64 x^{12} + 20 x^{11} - 28 x^{10} + 5 x^{9} - 28 x^{8} + 20 x^{7} + 64 x^{6} - 126 x^{5} + 110 x^{4} - 59 x^{3} + 22 x^{2} - 6 x + 1 \)

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:  \(-1878328153971890270208=-\,2^{12}\cdot 3^{9}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.20$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{15} a^{15} + \frac{2}{15} a^{14} - \frac{4}{15} a^{13} + \frac{4}{15} a^{12} - \frac{2}{15} a^{11} + \frac{4}{15} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{15} a^{6} + \frac{4}{15} a^{5} - \frac{2}{15} a^{4} - \frac{1}{15} a^{3} - \frac{4}{15} a^{2} + \frac{7}{15} a + \frac{2}{5}$, $\frac{1}{252675} a^{16} + \frac{2488}{252675} a^{15} - \frac{27557}{252675} a^{14} - \frac{758}{10107} a^{13} - \frac{101323}{252675} a^{12} - \frac{22583}{252675} a^{11} - \frac{7328}{16845} a^{10} + \frac{68188}{252675} a^{9} - \frac{10372}{84225} a^{8} - \frac{117107}{252675} a^{7} + \frac{7271}{16845} a^{6} - \frac{123653}{252675} a^{5} - \frac{84478}{252675} a^{4} + \frac{19793}{50535} a^{3} - \frac{94937}{252675} a^{2} + \frac{69868}{252675} a - \frac{16844}{252675}$, $\frac{1}{252675} a^{17} - \frac{632}{84225} a^{15} + \frac{11417}{84225} a^{14} + \frac{31882}{252675} a^{13} + \frac{84686}{252675} a^{12} + \frac{100649}{252675} a^{11} - \frac{81032}{252675} a^{10} - \frac{14111}{50535} a^{9} + \frac{31486}{252675} a^{8} - \frac{58}{225} a^{7} + \frac{47182}{252675} a^{6} + \frac{75556}{252675} a^{5} - \frac{80131}{252675} a^{4} + \frac{18371}{84225} a^{3} + \frac{1718}{84225} a^{2} - \frac{41981}{84225} a + \frac{31202}{252675}$

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

Class group and class number

Trivial group, which has 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:  \( -\frac{86702}{50535} a^{17} + \frac{450494}{50535} a^{16} - \frac{1533058}{50535} a^{15} + \frac{3825476}{50535} a^{14} - \frac{2093132}{16845} a^{13} + \frac{605877}{5615} a^{12} - \frac{572876}{50535} a^{11} - \frac{2472164}{50535} a^{10} + \frac{16874}{10107} a^{9} + \frac{2090}{10107} a^{8} + \frac{2682886}{50535} a^{7} + \frac{380824}{50535} a^{6} - \frac{1902184}{16845} a^{5} + \frac{2038748}{16845} a^{4} - \frac{3817324}{50535} a^{3} + \frac{1656092}{50535} a^{2} - \frac{554536}{50535} a + \frac{134683}{50535} \) (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:  \( 3594.02576505 \)
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

\(\Q(\sqrt{-3}) \), 3.1.2028.1 x3, 3.3.169.1, 6.0.12338352.2, 6.0.771147.1, 6.0.73008.1 x2, 9.3.8340725952.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.73008.1
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 R R ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$