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Properties

Label 18.0.69130423372...7888.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 13^{15}\cdot 61^{6}$
Root discriminant $66.75$
Ramified primes $2, 13, 61$
Class number $4448$ (GRH)
Class group $[2, 2224]$ (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![994969, -1826896, 3591174, -3603978, 3232214, -1492424, 552807, 135014, -53723, 25180, 34964, -8962, 1633, -784, 314, -46, 7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 7*x^16 - 46*x^15 + 314*x^14 - 784*x^13 + 1633*x^12 - 8962*x^11 + 34964*x^10 + 25180*x^9 - 53723*x^8 + 135014*x^7 + 552807*x^6 - 1492424*x^5 + 3232214*x^4 - 3603978*x^3 + 3591174*x^2 - 1826896*x + 994969)
 
gp: K = bnfinit(x^18 + 7*x^16 - 46*x^15 + 314*x^14 - 784*x^13 + 1633*x^12 - 8962*x^11 + 34964*x^10 + 25180*x^9 - 53723*x^8 + 135014*x^7 + 552807*x^6 - 1492424*x^5 + 3232214*x^4 - 3603978*x^3 + 3591174*x^2 - 1826896*x + 994969, 1)
 

Normalized defining polynomial

\( x^{18} + 7 x^{16} - 46 x^{15} + 314 x^{14} - 784 x^{13} + 1633 x^{12} - 8962 x^{11} + 34964 x^{10} + 25180 x^{9} - 53723 x^{8} + 135014 x^{7} + 552807 x^{6} - 1492424 x^{5} + 3232214 x^{4} - 3603978 x^{3} + 3591174 x^{2} - 1826896 x + 994969 \)

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:  \(-691304233720361896816117357477888=-\,2^{18}\cdot 13^{15}\cdot 61^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.75$
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:  $2, 13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{162244} a^{16} - \frac{4327}{40561} a^{15} - \frac{9057}{81122} a^{14} - \frac{8853}{81122} a^{13} - \frac{19598}{40561} a^{12} - \frac{15997}{81122} a^{11} + \frac{61521}{162244} a^{10} - \frac{11247}{40561} a^{9} + \frac{53695}{162244} a^{8} + \frac{2032}{40561} a^{7} - \frac{12719}{81122} a^{6} + \frac{35539}{81122} a^{5} - \frac{79803}{162244} a^{4} - \frac{26299}{81122} a^{3} + \frac{51321}{162244} a^{2} + \frac{17560}{40561} a + \frac{53401}{162244}$, $\frac{1}{815664053784538560433755025826893589917076115320559844} a^{17} - \frac{760708448099600124773458553374795401356189086725}{815664053784538560433755025826893589917076115320559844} a^{16} + \frac{22326619814897185167292393809645771721408550338467927}{203916013446134640108438756456723397479269028830139961} a^{15} + \frac{44805945607438049045336998091747383214106283734167085}{407832026892269280216877512913446794958538057660279922} a^{14} + \frac{2607100174618840036361954272696662604575893019298396}{203916013446134640108438756456723397479269028830139961} a^{13} + \frac{1996332772086423496020056963045501187339262673505379}{203916013446134640108438756456723397479269028830139961} a^{12} - \frac{26725522029572813104221438695541896971937917136230267}{815664053784538560433755025826893589917076115320559844} a^{11} - \frac{20523696892075047702384369637429691432389836660048655}{815664053784538560433755025826893589917076115320559844} a^{10} + \frac{222852676559483920365747042276856354380325743791316035}{815664053784538560433755025826893589917076115320559844} a^{9} - \frac{369808448216843701015124197753220292614755250499122879}{815664053784538560433755025826893589917076115320559844} a^{8} - \frac{1561697655835478357859783095606344461809755239572636}{203916013446134640108438756456723397479269028830139961} a^{7} + \frac{75937605309636089369810158880909481470843793044910243}{407832026892269280216877512913446794958538057660279922} a^{6} - \frac{95383729801633612539822272920412745015332812648508863}{815664053784538560433755025826893589917076115320559844} a^{5} + \frac{288225117405459897736159502166840757352724714291677151}{815664053784538560433755025826893589917076115320559844} a^{4} + \frac{31061188148175476464715304257293395018639419493774947}{815664053784538560433755025826893589917076115320559844} a^{3} - \frac{76252690109662269240386326427574407207375737030145185}{815664053784538560433755025826893589917076115320559844} a^{2} - \frac{323463270707350309870253183293746925283223966937392577}{815664053784538560433755025826893589917076115320559844} a + \frac{113173708123920303770526780860195640174254318369921489}{815664053784538560433755025826893589917076115320559844}$

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

Class group and class number

$C_{2}\times C_{2224}$, which has order $4448$ (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:  \( 230602.64598601736 \) (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

\(\Q(\sqrt{-13}) \), 3.3.169.1, 3.3.10309.1, 6.0.23762752.1, 6.0.88421200192.1, 9.9.1095593933629.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
13Data not computed
$61$61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
61.6.3.1$x^{6} - 122 x^{4} + 3721 x^{2} - 22698100$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
61.6.3.1$x^{6} - 122 x^{4} + 3721 x^{2} - 22698100$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$