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Properties

Label 18.0.88741447186...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 5^{9}\cdot 13^{9}$
Root discriminant $46.11$
Ramified primes $2, 3, 5, 13$
Class number $36$ (GRH)
Class group $[6, 6]$ (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![7020, 0, 1296, 0, -26640, 0, 35161, 0, -13950, 0, -1209, 0, 1540, 0, 351, 0, 30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 30*x^16 + 351*x^14 + 1540*x^12 - 1209*x^10 - 13950*x^8 + 35161*x^6 - 26640*x^4 + 1296*x^2 + 7020)
 
gp: K = bnfinit(x^18 + 30*x^16 + 351*x^14 + 1540*x^12 - 1209*x^10 - 13950*x^8 + 35161*x^6 - 26640*x^4 + 1296*x^2 + 7020, 1)
 

Normalized defining polynomial

\( x^{18} + 30 x^{16} + 351 x^{14} + 1540 x^{12} - 1209 x^{10} - 13950 x^{8} + 35161 x^{6} - 26640 x^{4} + 1296 x^{2} + 7020 \)

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:  \(-887414471860576333752000000000=-\,2^{12}\cdot 3^{21}\cdot 5^{9}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.11$
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:  $2, 3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{12} a^{8} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{9} + \frac{1}{4} a^{5} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{60} a^{10} + \frac{1}{20} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{180} a^{11} + \frac{1}{36} a^{9} - \frac{1}{15} a^{7} + \frac{1}{36} a^{5} + \frac{31}{180} a^{3} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{2880} a^{12} + \frac{1}{576} a^{10} + \frac{1}{960} a^{8} - \frac{17}{576} a^{6} - \frac{1}{2} a^{5} + \frac{77}{360} a^{4} - \frac{1}{2} a^{3} - \frac{5}{16} a^{2} - \frac{1}{2} a - \frac{3}{16}$, $\frac{1}{2880} a^{13} + \frac{1}{576} a^{11} + \frac{1}{960} a^{9} - \frac{17}{576} a^{7} - \frac{103}{360} a^{5} + \frac{3}{16} a^{3} - \frac{1}{2} a^{2} - \frac{3}{16} a$, $\frac{1}{74880} a^{14} + \frac{1}{37440} a^{12} - \frac{7}{2080} a^{10} + \frac{553}{37440} a^{8} - \frac{9929}{74880} a^{6} + \frac{1711}{6240} a^{4} + \frac{73}{312} a^{2} - \frac{1}{2} a + \frac{13}{32}$, $\frac{1}{149760} a^{15} - \frac{1}{149760} a^{14} - \frac{1}{6240} a^{13} - \frac{1}{74880} a^{12} - \frac{191}{74880} a^{11} + \frac{7}{4160} a^{10} + \frac{257}{37440} a^{9} + \frac{2567}{74880} a^{8} - \frac{2573}{49920} a^{7} - \frac{27511}{149760} a^{6} - \frac{8231}{37440} a^{5} - \frac{151}{12480} a^{4} + \frac{29}{1248} a^{3} - \frac{59}{208} a^{2} - \frac{13}{64} a - \frac{13}{64}$, $\frac{1}{61551360} a^{16} - \frac{59}{20517120} a^{14} - \frac{1}{5760} a^{13} - \frac{1}{18720} a^{12} - \frac{1}{1152} a^{11} + \frac{16427}{7693920} a^{10} + \frac{79}{1920} a^{9} - \frac{313259}{20517120} a^{8} + \frac{17}{1152} a^{7} + \frac{3357103}{20517120} a^{6} + \frac{193}{720} a^{5} - \frac{4970999}{15387840} a^{4} + \frac{47}{96} a^{3} - \frac{107459}{5129280} a^{2} + \frac{3}{32} a - \frac{11917}{26304}$, $\frac{1}{61551360} a^{17} - \frac{59}{20517120} a^{15} - \frac{1}{18720} a^{13} - \frac{1}{5760} a^{12} + \frac{16427}{7693920} a^{11} - \frac{1}{1152} a^{10} - \frac{313259}{20517120} a^{9} + \frac{79}{1920} a^{8} - \frac{1772177}{20517120} a^{7} + \frac{17}{1152} a^{6} - \frac{4970999}{15387840} a^{5} - \frac{167}{720} a^{4} - \frac{1389779}{5129280} a^{3} - \frac{1}{96} a^{2} - \frac{11917}{26304} a - \frac{13}{32}$

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

Class group and class number

$C_{6}\times C_{6}$, which has order $36$ (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:  \( 45058308.3492 \) (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

\(\Q(\sqrt{-195}) \), 3.1.1755.1 x3, 3.1.780.1 x3, 3.1.7020.2 x3, 3.1.7020.1 x3, 6.0.600604875.1, 6.0.118638000.1, 6.0.9609678000.2, 6.0.9609678000.1, 9.1.67459939560000.2 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 R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$$[\ ]_{3}^{2}$
$3$3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$