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Properties

Label 18.0.24962803242...2368.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 17^{12}$
Root discriminant $37.81$
Ramified primes $2, 3, 17$
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![2523, 10440, 16140, 7200, -5622, -2976, 4219, 1536, 2532, 864, -348, -120, 13, -132, 24, 0, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 6*x^16 + 24*x^14 - 132*x^13 + 13*x^12 - 120*x^11 - 348*x^10 + 864*x^9 + 2532*x^8 + 1536*x^7 + 4219*x^6 - 2976*x^5 - 5622*x^4 + 7200*x^3 + 16140*x^2 + 10440*x + 2523)
 
gp: K = bnfinit(x^18 + 6*x^16 + 24*x^14 - 132*x^13 + 13*x^12 - 120*x^11 - 348*x^10 + 864*x^9 + 2532*x^8 + 1536*x^7 + 4219*x^6 - 2976*x^5 - 5622*x^4 + 7200*x^3 + 16140*x^2 + 10440*x + 2523, 1)
 

Normalized defining polynomial

\( x^{18} + 6 x^{16} + 24 x^{14} - 132 x^{13} + 13 x^{12} - 120 x^{11} - 348 x^{10} + 864 x^{9} + 2532 x^{8} + 1536 x^{7} + 4219 x^{6} - 2976 x^{5} - 5622 x^{4} + 7200 x^{3} + 16140 x^{2} + 10440 x + 2523 \)

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:  \(-24962803242374579500132282368=-\,2^{12}\cdot 3^{21}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.81$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{10} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{3}{10} a^{8} + \frac{1}{5} a^{7} - \frac{3}{10} a^{5} + \frac{2}{5} a^{4} - \frac{1}{10} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{10} - \frac{1}{10} a^{9} - \frac{1}{5} a^{8} - \frac{1}{10} a^{7} - \frac{3}{10} a^{6} - \frac{1}{5} a^{5} + \frac{3}{10} a^{4} - \frac{1}{10} a^{3} + \frac{2}{5} a^{2} - \frac{1}{10} a + \frac{2}{5}$, $\frac{1}{10} a^{13} + \frac{1}{5} a^{10} + \frac{1}{10} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{3}{10} a^{6} - \frac{1}{5} a^{4} - \frac{1}{10} a^{3} - \frac{1}{5} a^{2} - \frac{1}{2} a - \frac{3}{10}$, $\frac{1}{10} a^{14} + \frac{1}{10} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{2} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{3}{10} a^{3} - \frac{3}{10} a^{2} + \frac{1}{10}$, $\frac{1}{340} a^{15} - \frac{3}{85} a^{14} - \frac{3}{68} a^{12} - \frac{7}{170} a^{11} - \frac{13}{170} a^{10} - \frac{33}{170} a^{9} - \frac{37}{170} a^{8} + \frac{1}{10} a^{7} + \frac{1}{85} a^{6} - \frac{9}{34} a^{5} + \frac{1}{34} a^{4} - \frac{99}{340} a^{3} - \frac{1}{34} a^{2} - \frac{49}{170} a - \frac{25}{68}$, $\frac{1}{4745227928540} a^{16} - \frac{3682763981}{4745227928540} a^{15} - \frac{8467598443}{1186306982135} a^{14} - \frac{141468650659}{4745227928540} a^{13} - \frac{155338897861}{4745227928540} a^{12} + \frac{3903031856}{237261396427} a^{11} + \frac{329052305821}{2372613964270} a^{10} - \frac{225527895291}{2372613964270} a^{9} - \frac{10653081986}{1186306982135} a^{8} - \frac{155947116599}{1186306982135} a^{7} - \frac{50495044843}{1186306982135} a^{6} + \frac{190918258136}{1186306982135} a^{5} + \frac{2066371308373}{4745227928540} a^{4} - \frac{1603825484733}{4745227928540} a^{3} + \frac{493910494333}{1186306982135} a^{2} - \frac{217317333281}{4745227928540} a - \frac{18180508019}{163628549260}$, $\frac{1}{7427301932169736100} a^{17} - \frac{8311}{168802316640221275} a^{16} - \frac{720798015270217}{1856825483042434025} a^{15} + \frac{370462674377011097}{7427301932169736100} a^{14} + \frac{34153318666707913}{3713650966084868050} a^{13} - \frac{66803483238935954}{1856825483042434025} a^{12} + \frac{78007944689598471}{3713650966084868050} a^{11} - \frac{650575616239836179}{3713650966084868050} a^{10} + \frac{24643125953253008}{109225028414260825} a^{9} + \frac{641121158294405907}{1856825483042434025} a^{8} - \frac{196685765803157299}{742730193216973610} a^{7} + \frac{3210787421872564}{1856825483042434025} a^{6} - \frac{51788131093964041}{135041853312177020} a^{5} + \frac{20103739163006447}{337604633280442550} a^{4} + \frac{600267459313334988}{1856825483042434025} a^{3} + \frac{1490984210664503667}{7427301932169736100} a^{2} + \frac{436713550457235763}{1856825483042434025} a + \frac{10381345381268687}{64028464932497725}$

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:  \( \frac{1222750509}{1867395407900} a^{17} - \frac{324045153}{933697703950} a^{16} + \frac{3885541669}{933697703950} a^{15} - \frac{4313647947}{1867395407900} a^{14} + \frac{731853796}{42440804725} a^{13} - \frac{44933378901}{466848851975} a^{12} + \frac{56834447929}{933697703950} a^{11} - \frac{111999544071}{933697703950} a^{10} - \frac{139578117417}{933697703950} a^{9} + \frac{296901944733}{466848851975} a^{8} + \frac{247370444569}{186739540790} a^{7} + \frac{351031277127}{933697703950} a^{6} + \frac{965174150681}{373479081580} a^{5} - \frac{1589418772851}{466848851975} a^{4} - \frac{890455147118}{466848851975} a^{3} + \frac{9391142641263}{1867395407900} a^{2} + \frac{682351786989}{84881609450} a + \frac{117266644651}{32196472550} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 11699771.704 \) (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

\(\Q(\sqrt{-3}) \), 3.1.31212.1 x3, 3.1.31212.2 x3, 3.1.108.1 x3, 3.1.867.1 x3, 6.0.2922566832.1, 6.0.2922566832.2, 6.0.34992.1, 6.0.2255067.2, 9.1.91219155960384.3 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$[\ ]_{3}^{2}$
$3$3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$17$17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$