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Properties

Label 18.0.16999975299...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 5^{9}\cdot 11^{9}\cdot 37^{14}$
Root discriminant $195.25$
Ramified primes $2, 5, 11, 37$
Class number $20998656$ (GRH)
Class group $[2, 2, 2, 4, 28, 23436]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![97992831511, -48726323693, 50489934410, -18290873776, 10743313753, -3039956271, 1307752076, -299009263, 103972952, -19494767, 5767404, -888043, 229872, -28695, 6531, -626, 120, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 120*x^16 - 626*x^15 + 6531*x^14 - 28695*x^13 + 229872*x^12 - 888043*x^11 + 5767404*x^10 - 19494767*x^9 + 103972952*x^8 - 299009263*x^7 + 1307752076*x^6 - 3039956271*x^5 + 10743313753*x^4 - 18290873776*x^3 + 50489934410*x^2 - 48726323693*x + 97992831511)
 
gp: K = bnfinit(x^18 - 7*x^17 + 120*x^16 - 626*x^15 + 6531*x^14 - 28695*x^13 + 229872*x^12 - 888043*x^11 + 5767404*x^10 - 19494767*x^9 + 103972952*x^8 - 299009263*x^7 + 1307752076*x^6 - 3039956271*x^5 + 10743313753*x^4 - 18290873776*x^3 + 50489934410*x^2 - 48726323693*x + 97992831511, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 120 x^{16} - 626 x^{15} + 6531 x^{14} - 28695 x^{13} + 229872 x^{12} - 888043 x^{11} + 5767404 x^{10} - 19494767 x^{9} + 103972952 x^{8} - 299009263 x^{7} + 1307752076 x^{6} - 3039956271 x^{5} + 10743313753 x^{4} - 18290873776 x^{3} + 50489934410 x^{2} - 48726323693 x + 97992831511 \)

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:  \(-169999752992335178942220108544792000000000=-\,2^{12}\cdot 5^{9}\cdot 11^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $195.25$
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:  $2, 5, 11, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{21573274919081748998194961798015577052472721320670866713023856027} a^{17} - \frac{3932192516589485500861436273377550641963780920764135919877134943}{21573274919081748998194961798015577052472721320670866713023856027} a^{16} + \frac{10775061304601677352119388435660517553640584984521203382980579599}{21573274919081748998194961798015577052472721320670866713023856027} a^{15} + \frac{7051280429078131559656947571140830200771807394931691406434373830}{21573274919081748998194961798015577052472721320670866713023856027} a^{14} + \frac{9782953007792609265057486225194679184986235888694592304656422336}{21573274919081748998194961798015577052472721320670866713023856027} a^{13} - \frac{2132473871184485960096633253660338760154842433156759280933385471}{21573274919081748998194961798015577052472721320670866713023856027} a^{12} + \frac{23341504783373219578584652648718159206084092192353898694425650}{209449271059046106778591862116656087888084673016222006922561709} a^{11} + \frac{4122175267799348938442180308088744944134221570255501692536389469}{21573274919081748998194961798015577052472721320670866713023856027} a^{10} + \frac{3854468970369002443112350858294501567993831097168777005106204130}{21573274919081748998194961798015577052472721320670866713023856027} a^{9} + \frac{10173543584642360774150414152207381993193746303128313047911673859}{21573274919081748998194961798015577052472721320670866713023856027} a^{8} - \frac{6724234453234574721542144550105436315243300780311305262270628996}{21573274919081748998194961798015577052472721320670866713023856027} a^{7} + \frac{4707889505555969417808794064549111374086876304539718893084575}{21573274919081748998194961798015577052472721320670866713023856027} a^{6} + \frac{7872188480275907162043682772496873855116911819472937190924126401}{21573274919081748998194961798015577052472721320670866713023856027} a^{5} - \frac{6979952140366853437061279911666786205464186963896717990925714653}{21573274919081748998194961798015577052472721320670866713023856027} a^{4} - \frac{1736844478173331294151999020685020316359366949095662510110549919}{21573274919081748998194961798015577052472721320670866713023856027} a^{3} - \frac{8683735471163407771373033985199345092701961798131186806789836101}{21573274919081748998194961798015577052472721320670866713023856027} a^{2} - \frac{6585499960571765961687200492272683545892986106909127004680858308}{21573274919081748998194961798015577052472721320670866713023856027} a - \frac{7207943175269073050327769631103364892094536016081162278482242034}{21573274919081748998194961798015577052472721320670866713023856027}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{28}\times C_{23436}$, which has order $20998656$ (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:  \( 615797.1340659427 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-55}) \), 3.3.1369.1, 3.3.148.1, 6.0.3644278000.2, 6.0.311813536375.2, 9.9.6075640136512.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$37.6.4.1$x^{6} + 518 x^{3} + 171125$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
37.12.10.1$x^{12} + 1998 x^{6} + 21390625$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$