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Properties

Label 18.0.66305754614...9472.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{27}\cdot 7^{12}\cdot 13^{12}\cdot 37^{12}$
Root discriminant $1852.95$
Ramified primes $2, 3, 7, 13, 37$
Class number $11945199357$ (GRH)
Class group $[3, 3, 3, 3, 3, 3, 9, 117, 15561]$ (GRH)
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![19780262567688384, 0, 0, 208357877056896, 0, 0, 951986143216, 0, 0, 2022824160, 0, 0, 1165924, 0, 0, -3024, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3024*x^15 + 1165924*x^12 + 2022824160*x^9 + 951986143216*x^6 + 208357877056896*x^3 + 19780262567688384)
 
gp: K = bnfinit(x^18 - 3024*x^15 + 1165924*x^12 + 2022824160*x^9 + 951986143216*x^6 + 208357877056896*x^3 + 19780262567688384, 1)
 

Normalized defining polynomial

\( x^{18} - 3024 x^{15} + 1165924 x^{12} + 2022824160 x^{9} + 951986143216 x^{6} + 208357877056896 x^{3} + 19780262567688384 \)

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:  \(-66305754614937078316581682973174263499455978739075060969472=-\,2^{12}\cdot 3^{27}\cdot 7^{12}\cdot 13^{12}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1852.95$
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:  $2, 3, 7, 13, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6}$, $\frac{1}{8} a^{7} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{592} a^{9} + \frac{21}{296} a^{6} - \frac{1}{148} a^{3} - \frac{27}{74}$, $\frac{1}{1184} a^{10} + \frac{21}{592} a^{7} + \frac{73}{296} a^{4} + \frac{47}{148} a$, $\frac{1}{1184} a^{11} + \frac{21}{592} a^{8} - \frac{1}{296} a^{5} - \frac{27}{148} a^{2}$, $\frac{1}{622784} a^{12} + \frac{15}{19462} a^{9} + \frac{317}{38924} a^{6} + \frac{8165}{38924} a^{3} + \frac{16657}{38924}$, $\frac{1}{19306304} a^{13} - \frac{1721}{4826576} a^{10} + \frac{897}{2413288} a^{7} + \frac{46291}{301661} a^{4} + \frac{427463}{1206644} a$, $\frac{1}{57918912} a^{14} - \frac{3865}{9653152} a^{11} + \frac{433903}{14479728} a^{8} - \frac{74947}{2413288} a^{5} - \frac{882847}{1809966} a^{2}$, $\frac{1}{144874599782598908742898560} a^{15} - \frac{1213426729590206389}{1857366663879473189011520} a^{12} + \frac{509613531617969835961}{905466248641243179643116} a^{9} + \frac{40048147158462403768037}{603644165760828786428744} a^{6} - \frac{11172556763040148608217}{532627205083084223319480} a^{3} - \frac{16543009153465710367087}{48680981109744256970060}$, $\frac{1}{315971502125848219968261759360} a^{16} + \frac{16271090863967398737}{1350305564640377008411375040} a^{13} - \frac{904640825710699819662415}{15798575106292410998413087968} a^{10} + \frac{49610779443390740815802139}{877698617016245055467393776} a^{7} + \frac{784632356130453561390001}{145207491785775836382473235} a^{4} + \frac{65950829858407739946054804}{274280817817576579833560555} a$, $\frac{1}{14242099486820482666849430541392640} a^{17} - \frac{125598077160055375169723}{60863673020600353277134318552960} a^{14} + \frac{36077727838251107539950914303}{178026243585256033335617881767408} a^{11} + \frac{165946764503689923730644822989}{19780693731695114815068653529712} a^{8} - \frac{1006373664021961566481047163897}{52360659878016480392828788755120} a^{5} - \frac{18768739546514270282621298447639}{49451734329237787037671633824280} a^{2}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{9}\times C_{117}\times C_{15561}$, which has order $11945199357$ (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:  \( -\frac{1951}{40021838888028960} a^{15} + \frac{4314443}{26681225925352640} a^{12} - \frac{11140981}{108167132129808} a^{9} - \frac{48997265807}{667030648133816} a^{6} - \frac{3043881143243}{147139113558930} a^{3} - \frac{81742243740317}{53792794204340} \) (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:  \( 87969966278262.44 \) (assuming GRH)
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

\(\Q(\sqrt{-3}) \), Deg 3 x3, Deg 3, 6.0.55520863552535472.1, Deg 6 x2, 6.0.2529669345612397443.1, Deg 9 x3

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

Sibling fields

Degree 6 sibling: data not computed
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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$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$$[\ ]_{3}^{2}$
$3$3.6.9.11$x^{6} + 6 x^{4} + 12$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.11$x^{6} + 6 x^{4} + 12$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.11$x^{6} + 6 x^{4} + 12$$6$$1$$9$$C_6$$[2]_{2}$
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
$13$13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
$37$37.3.2.2$x^{3} + 74$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.2$x^{3} + 74$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.2$x^{3} + 74$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.2$x^{3} + 74$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.2$x^{3} + 74$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.2$x^{3} + 74$$3$$1$$2$$C_3$$[\ ]_{3}$