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Properties

Label 18.0.15796345197...7184.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 7^{9}\cdot 13^{9}\cdot 37^{14}$
Root discriminant $251.14$
Ramified primes $2, 7, 13, 37$
Class number $229709088$ (GRH)
Class group $[2, 2, 2, 28713636]$ (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![4931895139903, -1522944032030, 1622269413029, -392285922997, 231046792381, -45165879675, 19191974213, -3077994253, 1044507053, -137757260, 39236496, -4210681, 1030359, -87177, 18375, -1130, 201, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 201*x^16 - 1130*x^15 + 18375*x^14 - 87177*x^13 + 1030359*x^12 - 4210681*x^11 + 39236496*x^10 - 137757260*x^9 + 1044507053*x^8 - 3077994253*x^7 + 19191974213*x^6 - 45165879675*x^5 + 231046792381*x^4 - 392285922997*x^3 + 1622269413029*x^2 - 1522944032030*x + 4931895139903)
 
gp: K = bnfinit(x^18 - 7*x^17 + 201*x^16 - 1130*x^15 + 18375*x^14 - 87177*x^13 + 1030359*x^12 - 4210681*x^11 + 39236496*x^10 - 137757260*x^9 + 1044507053*x^8 - 3077994253*x^7 + 19191974213*x^6 - 45165879675*x^5 + 231046792381*x^4 - 392285922997*x^3 + 1622269413029*x^2 - 1522944032030*x + 4931895139903, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 201 x^{16} - 1130 x^{15} + 18375 x^{14} - 87177 x^{13} + 1030359 x^{12} - 4210681 x^{11} + 39236496 x^{10} - 137757260 x^{9} + 1044507053 x^{8} - 3077994253 x^{7} + 19191974213 x^{6} - 45165879675 x^{5} + 231046792381 x^{4} - 392285922997 x^{3} + 1622269413029 x^{2} - 1522944032030 x + 4931895139903 \)

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:  \(-15796345197168827671044286969235239338717184=-\,2^{12}\cdot 7^{9}\cdot 13^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $251.14$
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:  $2, 7, 13, 37$
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}{388502166248810523783828249243144205826625118021624891802283123998266879} a^{17} - \frac{120195591851612495431716636183222841041894641272599112449274080029172350}{388502166248810523783828249243144205826625118021624891802283123998266879} a^{16} - \frac{193574962717704346074027696060095379090627521735344163045303789009789908}{388502166248810523783828249243144205826625118021624891802283123998266879} a^{15} - \frac{120790758886008308533853382924760385830634961771319528318075930342583410}{388502166248810523783828249243144205826625118021624891802283123998266879} a^{14} + \frac{92858687211173248728084109203368333474618783599933181994156603402116540}{388502166248810523783828249243144205826625118021624891802283123998266879} a^{13} + \frac{141491408105985621839617075141391079051121575950477133946876728104610961}{388502166248810523783828249243144205826625118021624891802283123998266879} a^{12} - \frac{51933038720097197098882265888487240915270790135891244778042134924684944}{388502166248810523783828249243144205826625118021624891802283123998266879} a^{11} + \frac{149340978688954165747292414588868385369064065598964493867910342839974799}{388502166248810523783828249243144205826625118021624891802283123998266879} a^{10} + \frac{12024278479947990165013662299863978280029327243176194528130878096252954}{388502166248810523783828249243144205826625118021624891802283123998266879} a^{9} - \frac{172264925983845228344161256744689853912853658613304059271194442013248766}{388502166248810523783828249243144205826625118021624891802283123998266879} a^{8} + \frac{71564493297597248315419963687310091153009419608339261317085116878880334}{388502166248810523783828249243144205826625118021624891802283123998266879} a^{7} - \frac{67092662050508303302464473207753240028702647638712854459400675484491630}{388502166248810523783828249243144205826625118021624891802283123998266879} a^{6} + \frac{164792821555735341838314687850871111443014698817599405293993188897875858}{388502166248810523783828249243144205826625118021624891802283123998266879} a^{5} - \frac{39426934775170913555687458308774532016937423782736279934654654573735550}{388502166248810523783828249243144205826625118021624891802283123998266879} a^{4} + \frac{77460759169155016060962658665292397163682401800308350493214313516004264}{388502166248810523783828249243144205826625118021624891802283123998266879} a^{3} + \frac{55904014149048683583841491376539588367815335700483477090228726713365590}{388502166248810523783828249243144205826625118021624891802283123998266879} a^{2} - \frac{65368181994673385382929606335043302577756518661277563667531093106538550}{388502166248810523783828249243144205826625118021624891802283123998266879} a + \frac{33251568463155613138154157343576115966555027701395780372604050203904139}{388502166248810523783828249243144205826625118021624891802283123998266879}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{28713636}$, which has order $229709088$ (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

\(\Q(\sqrt{-91}) \), 3.3.1369.1, 3.3.148.1, 6.0.16506219184.7, 6.0.1412313378931.4, 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
2Data not computed
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.12.6.1$x^{12} + 338 x^{8} + 8788 x^{6} + 28561 x^{4} + 19307236$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$37$37.6.4.1$x^{6} + 518 x^{3} + 171125$$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}$