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Properties

Label 18.0.19451929844...1627.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 7^{15}\cdot 29^{9}$
Root discriminant $117.93$
Ramified primes $3, 7, 29$
Class number $1870128$ (GRH)
Class group $[2, 18, 51948]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![105164497817, -40252088898, 49541028015, -16134045101, 11675224038, -3151773927, 1588480509, -340709811, 136802574, -22276859, 7949448, -929568, 317736, -24849, 8433, -397, 135, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 135*x^16 - 397*x^15 + 8433*x^14 - 24849*x^13 + 317736*x^12 - 929568*x^11 + 7949448*x^10 - 22276859*x^9 + 136802574*x^8 - 340709811*x^7 + 1588480509*x^6 - 3151773927*x^5 + 11675224038*x^4 - 16134045101*x^3 + 49541028015*x^2 - 40252088898*x + 105164497817)
 
gp: K = bnfinit(x^18 - 3*x^17 + 135*x^16 - 397*x^15 + 8433*x^14 - 24849*x^13 + 317736*x^12 - 929568*x^11 + 7949448*x^10 - 22276859*x^9 + 136802574*x^8 - 340709811*x^7 + 1588480509*x^6 - 3151773927*x^5 + 11675224038*x^4 - 16134045101*x^3 + 49541028015*x^2 - 40252088898*x + 105164497817, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 135 x^{16} - 397 x^{15} + 8433 x^{14} - 24849 x^{13} + 317736 x^{12} - 929568 x^{11} + 7949448 x^{10} - 22276859 x^{9} + 136802574 x^{8} - 340709811 x^{7} + 1588480509 x^{6} - 3151773927 x^{5} + 11675224038 x^{4} - 16134045101 x^{3} + 49541028015 x^{2} - 40252088898 x + 105164497817 \)

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:  \(-19451929844843151808873875967466301627=-\,3^{24}\cdot 7^{15}\cdot 29^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $117.93$
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:  $3, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1827=3^{2}\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1827}(1,·)$, $\chi_{1827}(1219,·)$, $\chi_{1827}(202,·)$, $\chi_{1827}(1420,·)$, $\chi_{1827}(724,·)$, $\chi_{1827}(1045,·)$, $\chi_{1827}(88,·)$, $\chi_{1827}(985,·)$, $\chi_{1827}(1306,·)$, $\chi_{1827}(610,·)$, $\chi_{1827}(811,·)$, $\chi_{1827}(115,·)$, $\chi_{1827}(436,·)$, $\chi_{1827}(1333,·)$, $\chi_{1827}(1654,·)$, $\chi_{1827}(376,·)$, $\chi_{1827}(697,·)$, $\chi_{1827}(1594,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{673} a^{12} - \frac{258}{673} a^{11} + \frac{51}{673} a^{10} + \frac{145}{673} a^{9} + \frac{141}{673} a^{8} - \frac{240}{673} a^{7} - \frac{102}{673} a^{6} - \frac{24}{673} a^{5} + \frac{327}{673} a^{4} + \frac{286}{673} a^{3} + \frac{42}{673} a^{2} - \frac{46}{673} a + \frac{111}{673}$, $\frac{1}{673} a^{13} + \frac{114}{673} a^{11} - \frac{157}{673} a^{10} - \frac{137}{673} a^{9} - \frac{204}{673} a^{8} - \frac{106}{673} a^{7} - \frac{93}{673} a^{6} + \frac{192}{673} a^{5} - \frac{146}{673} a^{4} - \frac{200}{673} a^{3} + \frac{22}{673} a^{2} - \frac{316}{673} a - \frac{301}{673}$, $\frac{1}{673} a^{14} + \frac{316}{673} a^{11} + \frac{106}{673} a^{10} + \frac{91}{673} a^{9} - \frac{28}{673} a^{8} - \frac{326}{673} a^{7} - \frac{294}{673} a^{6} - \frac{102}{673} a^{5} + \frac{210}{673} a^{4} - \frac{278}{673} a^{3} + \frac{280}{673} a^{2} + \frac{232}{673} a + \frac{133}{673}$, $\frac{1}{673} a^{15} + \frac{201}{673} a^{11} + \frac{127}{673} a^{10} - \frac{84}{673} a^{9} + \frac{209}{673} a^{8} + \frac{170}{673} a^{7} - \frac{174}{673} a^{6} - \frac{282}{673} a^{5} + \frac{32}{673} a^{4} + \frac{86}{673} a^{3} - \frac{253}{673} a^{2} - \frac{137}{673} a - \frac{80}{673}$, $\frac{1}{47783} a^{16} - \frac{16}{47783} a^{15} - \frac{22}{47783} a^{14} - \frac{22}{47783} a^{13} - \frac{7}{47783} a^{12} - \frac{611}{47783} a^{11} + \frac{22721}{47783} a^{10} + \frac{3363}{47783} a^{9} + \frac{19039}{47783} a^{8} - \frac{8751}{47783} a^{7} + \frac{13388}{47783} a^{6} + \frac{21689}{47783} a^{5} + \frac{14275}{47783} a^{4} + \frac{9969}{47783} a^{3} + \frac{9394}{47783} a^{2} - \frac{2624}{47783} a + \frac{732}{47783}$, $\frac{1}{15059070084564531268168574244564034543701109262342045400013184370181} a^{17} - \frac{56485277526700154967369868704373368954232603105951168028399708}{15059070084564531268168574244564034543701109262342045400013184370181} a^{16} - \frac{1934875981724258816612118608249945553856094416407115483096873493}{15059070084564531268168574244564034543701109262342045400013184370181} a^{15} + \frac{8105408572385304111717645380495409030575322584874472596542069460}{15059070084564531268168574244564034543701109262342045400013184370181} a^{14} + \frac{5706105040872743235897276197579566468778644295696934444708794353}{15059070084564531268168574244564034543701109262342045400013184370181} a^{13} + \frac{3619085352778817807591503628622953471841947386370194844113472841}{15059070084564531268168574244564034543701109262342045400013184370181} a^{12} + \frac{2360612478632868230292830189821957423509336366831741981355321314336}{15059070084564531268168574244564034543701109262342045400013184370181} a^{11} + \frac{3168398630517582520042212684171581282948493304818158386190867864372}{15059070084564531268168574244564034543701109262342045400013184370181} a^{10} - \frac{6530009453265928324198567528541131604171661773246475282890470240461}{15059070084564531268168574244564034543701109262342045400013184370181} a^{9} - \frac{6644751131024374221626150789315054557171399036628907484335383235963}{15059070084564531268168574244564034543701109262342045400013184370181} a^{8} + \frac{3703387235701569064552947972909539324800472807931752452059120507579}{15059070084564531268168574244564034543701109262342045400013184370181} a^{7} + \frac{3203724553100512945782674456249767986278630716831775701285469211854}{15059070084564531268168574244564034543701109262342045400013184370181} a^{6} - \frac{6861465237574458073644759708130357827621615859376965064968081547561}{15059070084564531268168574244564034543701109262342045400013184370181} a^{5} - \frac{6615085671441351786433497445612249848683957625596573221216661142991}{15059070084564531268168574244564034543701109262342045400013184370181} a^{4} - \frac{71180097111855320442143781674113805464394552811738033892461852254}{212099578655838468565754566824845556953536750173831625352298371411} a^{3} - \frac{6584372278536511315334232059701476092219677707417776380288338081230}{15059070084564531268168574244564034543701109262342045400013184370181} a^{2} - \frac{6532512396178871259235946643794183384870188501168219657486588028512}{15059070084564531268168574244564034543701109262342045400013184370181} a - \frac{4633619116548272135716587072082648643431850965661836884488032907657}{15059070084564531268168574244564034543701109262342045400013184370181}$

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

Class group and class number

$C_{2}\times C_{18}\times C_{51948}$, which has order $1870128$ (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:  \( 54408.48888868202 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-203}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, 6.0.54885566547.7, 6.0.2689392760803.4, 6.0.409905923.1, 6.0.2689392760803.3, 9.9.62523502209.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
7Data not computed
$29$29.6.3.1$x^{6} - 58 x^{4} + 841 x^{2} - 219501$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.6.3.1$x^{6} - 58 x^{4} + 841 x^{2} - 219501$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.6.3.1$x^{6} - 58 x^{4} + 841 x^{2} - 219501$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$