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Properties

Label 18.0.85756462130...3888.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{30}\cdot 7^{9}\cdot 13^{14}$
Root discriminant $242.76$
Ramified primes $2, 3, 7, 13$
Class number $66433500$ (GRH)
Class group $[5, 30, 442890]$ (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![107610112, 761856, 42863616, 12964608, 5211552, 3729264, 6179676, 613290, -1018584, 315807, 339819, -26646, -41898, -12, 2664, 108, -78, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 78*x^16 + 108*x^15 + 2664*x^14 - 12*x^13 - 41898*x^12 - 26646*x^11 + 339819*x^10 + 315807*x^9 - 1018584*x^8 + 613290*x^7 + 6179676*x^6 + 3729264*x^5 + 5211552*x^4 + 12964608*x^3 + 42863616*x^2 + 761856*x + 107610112)
 
gp: K = bnfinit(x^18 - 3*x^17 - 78*x^16 + 108*x^15 + 2664*x^14 - 12*x^13 - 41898*x^12 - 26646*x^11 + 339819*x^10 + 315807*x^9 - 1018584*x^8 + 613290*x^7 + 6179676*x^6 + 3729264*x^5 + 5211552*x^4 + 12964608*x^3 + 42863616*x^2 + 761856*x + 107610112, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 78 x^{16} + 108 x^{15} + 2664 x^{14} - 12 x^{13} - 41898 x^{12} - 26646 x^{11} + 339819 x^{10} + 315807 x^{9} - 1018584 x^{8} + 613290 x^{7} + 6179676 x^{6} + 3729264 x^{5} + 5211552 x^{4} + 12964608 x^{3} + 42863616 x^{2} + 761856 x + 107610112 \)

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:  \(-8575646213021703940521028812918699203493888=-\,2^{18}\cdot 3^{30}\cdot 7^{9}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $242.76$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{10} + \frac{1}{32} a^{9} - \frac{1}{16} a^{7} - \frac{1}{32} a^{6} + \frac{7}{32} a^{5} - \frac{1}{8} a^{4} + \frac{5}{16} a^{3} - \frac{3}{8} a^{2}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{9} - \frac{1}{16} a^{8} + \frac{1}{32} a^{7} - \frac{3}{32} a^{5} + \frac{3}{16} a^{4} + \frac{1}{16} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{12} - \frac{1}{64} a^{10} - \frac{1}{32} a^{9} + \frac{1}{64} a^{8} - \frac{1}{8} a^{7} + \frac{5}{64} a^{6} - \frac{1}{32} a^{5} - \frac{3}{32} a^{4} - \frac{5}{16} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{128} a^{13} - \frac{1}{128} a^{12} - \frac{1}{128} a^{11} - \frac{1}{128} a^{10} + \frac{3}{128} a^{9} + \frac{7}{128} a^{8} - \frac{3}{128} a^{7} - \frac{7}{128} a^{6} - \frac{1}{32} a^{5} + \frac{1}{64} a^{4} - \frac{7}{32} a^{3} + \frac{1}{4} a$, $\frac{1}{512} a^{14} + \frac{1}{512} a^{13} - \frac{1}{512} a^{12} - \frac{3}{512} a^{11} - \frac{1}{512} a^{10} - \frac{23}{512} a^{9} - \frac{19}{512} a^{8} - \frac{29}{512} a^{7} - \frac{5}{64} a^{6} - \frac{37}{256} a^{5} - \frac{9}{128} a^{4} + \frac{3}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{111616} a^{15} - \frac{39}{55808} a^{14} - \frac{57}{27904} a^{13} - \frac{5}{13952} a^{12} + \frac{7}{1744} a^{11} + \frac{317}{27904} a^{10} - \frac{259}{55808} a^{9} + \frac{7}{256} a^{8} + \frac{9159}{111616} a^{7} + \frac{2409}{55808} a^{6} + \frac{377}{55808} a^{5} + \frac{2105}{27904} a^{4} - \frac{217}{6976} a^{3} + \frac{1303}{3488} a^{2} - \frac{193}{436} a + \frac{43}{109}$, $\frac{1}{15514624} a^{16} + \frac{31}{15514624} a^{15} + \frac{597}{3878656} a^{14} - \frac{18223}{7757312} a^{13} + \frac{18645}{7757312} a^{12} - \frac{32611}{7757312} a^{11} - \frac{5169}{969664} a^{10} + \frac{151}{35584} a^{9} - \frac{682119}{15514624} a^{8} + \frac{755719}{15514624} a^{7} - \frac{270949}{3878656} a^{6} - \frac{1759519}{7757312} a^{5} + \frac{90147}{3878656} a^{4} + \frac{340833}{969664} a^{3} - \frac{166243}{484832} a^{2} - \frac{4099}{15151} a - \frac{16}{139}$, $\frac{1}{2331915440326290428610091720544486973104128} a^{17} + \frac{47231202009781093524138114458308365}{2331915440326290428610091720544486973104128} a^{16} + \frac{4803122162036200129449975394614388765}{1165957720163145214305045860272243486552064} a^{15} + \frac{146954354070255834025628434297568422899}{582978860081572607152522930136121743276032} a^{14} + \frac{431264930315035747554569519542769356027}{291489430040786303576261465068060871638016} a^{13} - \frac{758982348874490130747520804323689304311}{582978860081572607152522930136121743276032} a^{12} + \frac{4013634816956305117845734421279262645939}{1165957720163145214305045860272243486552064} a^{11} + \frac{9647721978890140923275360683732932988173}{1165957720163145214305045860272243486552064} a^{10} + \frac{63573837451429681469068851288196344933195}{2331915440326290428610091720544486973104128} a^{9} - \frac{67125287826721343422430076608938591935553}{2331915440326290428610091720544486973104128} a^{8} - \frac{10702559815446220741620486955892772447819}{145744715020393151788130732534030435819008} a^{7} + \frac{83289437704830566476344917291242186871917}{1165957720163145214305045860272243486552064} a^{6} + \frac{62677253319508541621712517773495059089011}{582978860081572607152522930136121743276032} a^{5} + \frac{2423669879869172997361757447237587324581}{145744715020393151788130732534030435819008} a^{4} - \frac{7889338063323892755953300216533647610831}{72872357510196575894065366267015217909504} a^{3} + \frac{594698146618047490069505441277696681523}{2277261172193642996689542695844225559672} a^{2} - \frac{678613188205048270579452403240481844443}{2277261172193642996689542695844225559672} a + \frac{135116986084013519478713036102026536360}{284657646524205374586192836980528194959}$

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

Class group and class number

$C_{5}\times C_{30}\times C_{442890}$, which has order $66433500$ (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:  \( 3533133948.6916637 \) (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

\(\Q(\sqrt{-7}) \), 3.3.13689.1, 3.3.2808.1, 6.0.64274331303.4, 6.0.2704508352.7, 9.9.460990789028310528.1

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
3Data not computed
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.6.4.1$x^{6} + 39 x^{3} + 676$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.12.10.2$x^{12} + 39 x^{6} + 676$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$