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Properties

Label 18.0.58141490556...2368.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{24}\cdot 43^{9}$
Root discriminant $45.04$
Ramified primes $2, 3, 43$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2107, 0, -19368, 0, 35820, 0, 39119, 0, 14616, 0, -6168, 0, 101, 0, 180, 0, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 24*x^16 + 180*x^14 + 101*x^12 - 6168*x^10 + 14616*x^8 + 39119*x^6 + 35820*x^4 - 19368*x^2 + 2107)
 
gp: K = bnfinit(x^18 - 24*x^16 + 180*x^14 + 101*x^12 - 6168*x^10 + 14616*x^8 + 39119*x^6 + 35820*x^4 - 19368*x^2 + 2107, 1)
 

Normalized defining polynomial

\( x^{18} - 24 x^{16} + 180 x^{14} + 101 x^{12} - 6168 x^{10} + 14616 x^{8} + 39119 x^{6} + 35820 x^{4} - 19368 x^{2} + 2107 \)

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:  \(-581414905561488081454979002368=-\,2^{12}\cdot 3^{24}\cdot 43^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.04$
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:  $2, 3, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{36} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{5}{36}$, $\frac{1}{36} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{5}{36} a$, $\frac{1}{85140} a^{14} + \frac{569}{85140} a^{12} - \frac{969}{4730} a^{10} + \frac{204}{2365} a^{8} - \frac{1}{2} a^{7} + \frac{4027}{14190} a^{6} - \frac{1}{2} a^{5} - \frac{6649}{14190} a^{4} - \frac{1}{2} a^{3} + \frac{6989}{85140} a^{2} - \frac{1}{2} a - \frac{959}{1980}$, $\frac{1}{595980} a^{15} + \frac{1916}{148995} a^{13} + \frac{5157}{66220} a^{11} - \frac{1}{4} a^{10} + \frac{7911}{66220} a^{9} - \frac{1}{4} a^{8} + \frac{15149}{198660} a^{7} + \frac{1}{4} a^{6} - \frac{62963}{198660} a^{5} - \frac{1}{4} a^{4} + \frac{14137}{297990} a^{3} + \frac{1}{4} a^{2} + \frac{2341}{13860} a + \frac{1}{4}$, $\frac{1}{3797409341880} a^{16} + \frac{4682131}{3797409341880} a^{14} - \frac{48542760599}{3797409341880} a^{12} - \frac{6798883345}{42193437132} a^{10} - \frac{24801037585}{126580311396} a^{8} + \frac{29008411463}{126580311396} a^{6} - \frac{1}{2} a^{5} + \frac{3624148859}{11904104520} a^{4} - \frac{1690162363369}{3797409341880} a^{2} - \frac{6222216019}{12615977880}$, $\frac{1}{3797409341880} a^{17} - \frac{337915}{759481868376} a^{15} + \frac{8108077447}{3797409341880} a^{13} + \frac{53905483}{4906213620} a^{11} - \frac{5912806847}{90414508140} a^{9} + \frac{255004959463}{632901556980} a^{7} - \frac{1}{2} a^{6} + \frac{90591355}{2380820904} a^{5} - \frac{1}{2} a^{4} + \frac{977741027597}{3797409341880} a^{3} - \frac{24128180539}{88311845160} a - \frac{1}{2}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (assuming GRH)

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:  \( -1 \) (order $2$)
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 11837492.6514 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-43}) \), 3.1.13932.1 x3, 3.1.13932.2 x3, 3.1.3483.1 x3, 3.1.172.1 x3, 6.0.8346326832.1, 6.0.8346326832.2, 6.0.521645427.1, 6.0.1272112.1, 9.1.116281025423424.4 x9

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

Sibling fields

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{18}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$