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Properties

Label 18.0.33598511125...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{30}\cdot 3^{6}\cdot 5^{6}\cdot 7^{14}\cdot 181^{14}$
Root discriminant $2027.77$
Ramified primes $2, 3, 5, 7, 181$
Class number $97477447680$ (GRH)
Class group $[2, 6, 12, 12, 12, 12, 391740]$ (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![1286936844705604, 0, 561033372337329, 0, 88881650645340, 0, 6554580709188, 0, 244755477132, 0, 4846114662, 0, 52656708, 0, 310692, 0, 912, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 912*x^16 + 310692*x^14 + 52656708*x^12 + 4846114662*x^10 + 244755477132*x^8 + 6554580709188*x^6 + 88881650645340*x^4 + 561033372337329*x^2 + 1286936844705604)
 
gp: K = bnfinit(x^18 + 912*x^16 + 310692*x^14 + 52656708*x^12 + 4846114662*x^10 + 244755477132*x^8 + 6554580709188*x^6 + 88881650645340*x^4 + 561033372337329*x^2 + 1286936844705604, 1)
 

Normalized defining polynomial

\( x^{18} + 912 x^{16} + 310692 x^{14} + 52656708 x^{12} + 4846114662 x^{10} + 244755477132 x^{8} + 6554580709188 x^{6} + 88881650645340 x^{4} + 561033372337329 x^{2} + 1286936844705604 \)

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:  \(-335985111257225502524094185723848201140371850113581056000000=-\,2^{30}\cdot 3^{6}\cdot 5^{6}\cdot 7^{14}\cdot 181^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2027.77$
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:  $2, 3, 5, 7, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{4} - \frac{1}{6} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{5} - \frac{1}{6} a^{3} - \frac{1}{3} a$, $\frac{1}{36} a^{6} + \frac{1}{12} a^{2} + \frac{1}{9}$, $\frac{1}{72} a^{7} - \frac{1}{72} a^{6} + \frac{1}{24} a^{3} - \frac{1}{24} a^{2} + \frac{1}{18} a - \frac{1}{18}$, $\frac{1}{432} a^{8} - \frac{5}{432} a^{6} + \frac{1}{144} a^{4} + \frac{205}{432} a^{2} + \frac{49}{108}$, $\frac{1}{864} a^{9} - \frac{1}{864} a^{8} - \frac{5}{864} a^{7} + \frac{5}{864} a^{6} + \frac{1}{288} a^{5} - \frac{1}{288} a^{4} + \frac{205}{864} a^{3} - \frac{205}{864} a^{2} + \frac{49}{216} a - \frac{49}{216}$, $\frac{1}{10368} a^{10} - \frac{1}{2592} a^{8} - \frac{1}{5184} a^{6} + \frac{13}{648} a^{4} - \frac{1327}{10368} a^{2} - \frac{383}{2592}$, $\frac{1}{10368} a^{11} - \frac{1}{2592} a^{9} - \frac{1}{5184} a^{7} + \frac{13}{648} a^{5} - \frac{1327}{10368} a^{3} - \frac{383}{2592} a$, $\frac{1}{1244160} a^{12} - \frac{1}{46080} a^{10} - \frac{11}{69120} a^{8} - \frac{6929}{622080} a^{6} - \frac{4231}{138240} a^{4} - \frac{5171}{15360} a^{2} + \frac{88009}{311040}$, $\frac{1}{2488320} a^{13} - \frac{1}{2488320} a^{12} - \frac{1}{92160} a^{11} + \frac{1}{92160} a^{10} - \frac{11}{138240} a^{9} + \frac{11}{138240} a^{8} - \frac{6929}{1244160} a^{7} + \frac{6929}{1244160} a^{6} - \frac{4231}{276480} a^{5} + \frac{4231}{276480} a^{4} - \frac{5171}{30720} a^{3} + \frac{5171}{30720} a^{2} + \frac{88009}{622080} a - \frac{88009}{622080}$, $\frac{1}{2597806080} a^{14} - \frac{289}{1298903040} a^{12} + \frac{679}{32071680} a^{10} - \frac{47443}{64945152} a^{8} + \frac{26421359}{2597806080} a^{6} - \frac{285301}{16035840} a^{4} + \frac{110069497}{2597806080} a^{2} - \frac{213614879}{649451520}$, $\frac{1}{2597806080} a^{15} + \frac{233}{1298903040} a^{13} - \frac{1}{2488320} a^{12} + \frac{331}{32071680} a^{11} + \frac{1}{92160} a^{10} + \frac{56393}{162362880} a^{9} - \frac{149}{138240} a^{8} - \frac{3079993}{2597806080} a^{7} + \frac{14129}{1244160} a^{6} - \frac{475019}{16035840} a^{5} + \frac{3271}{276480} a^{4} + \frac{289166653}{2597806080} a^{3} - \frac{19061}{276480} a^{2} + \frac{25595797}{649451520} a - \frac{229129}{622080}$, $\frac{1}{141068459381568332651374510080} a^{16} + \frac{1264394495387404987}{28213691876313666530274902016} a^{14} + \frac{29367950730950958082981}{141068459381568332651374510080} a^{12} - \frac{665597551386787808765333}{28213691876313666530274902016} a^{10} - \frac{36590674156956790956513929}{141068459381568332651374510080} a^{8} - \frac{1735465808229335600362586563}{141068459381568332651374510080} a^{6} - \frac{8573312321007223512250330433}{141068459381568332651374510080} a^{4} - \frac{51061997903851061460750062971}{141068459381568332651374510080} a^{2} + \frac{12443942321228678888005525417}{35267114845392083162843627520}$, $\frac{1}{2530338043572681485919404669953966080} a^{17} + \frac{23380884315205427194796251}{506067608714536297183880933990793216} a^{15} + \frac{59759862862517026996772848481}{506067608714536297183880933990793216} a^{13} - \frac{31355309123658332988856784859913}{2530338043572681485919404669953966080} a^{11} + \frac{336120165517580564235387785296759}{2530338043572681485919404669953966080} a^{9} - \frac{290838083869960959844234032858695}{506067608714536297183880933990793216} a^{7} - \frac{1}{72} a^{6} + \frac{65865273266849938910149396557216511}{2530338043572681485919404669953966080} a^{5} + \frac{50912899968473897008794740334358561}{506067608714536297183880933990793216} a^{3} - \frac{1}{24} a^{2} + \frac{104554190977020607290303039164634313}{632584510893170371479851167488491520} a - \frac{1}{18}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{12}\times C_{12}\times C_{12}\times C_{12}\times C_{391740}$, which has order $97477447680$ (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:  \( -\frac{371321798780999240923}{84344601452422716197313488998465536} a^{17} - \frac{330470330984865867987605}{84344601452422716197313488998465536} a^{15} - \frac{108068758921284805072592503}{84344601452422716197313488998465536} a^{13} - \frac{17162650035442160333699334389}{84344601452422716197313488998465536} a^{11} - \frac{1422033131736216052212722964493}{84344601452422716197313488998465536} a^{9} - \frac{60276401057003079711257546857079}{84344601452422716197313488998465536} a^{7} - \frac{1210479919969556152042736390250997}{84344601452422716197313488998465536} a^{5} - \frac{12135901422384688097178496101548927}{84344601452422716197313488998465536} a^{3} - \frac{18726855528572098448046148010171659}{21086150363105679049328372249616384} a \) (order $4$)
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:  \( 49926751662940.16 \) (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

\(\Q(\sqrt{-1}) \), 3.3.1605289.2, 3.3.152040.1, 6.0.164924977505344.1, 6.0.369858585600.1, 9.9.9056909796008436008506944000.1

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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$181$181.3.2.1$x^{3} - 181$$3$$1$$2$$C_3$$[\ ]_{3}$
181.3.2.1$x^{3} - 181$$3$$1$$2$$C_3$$[\ ]_{3}$
181.6.5.1$x^{6} - 181$$6$$1$$5$$C_6$$[\ ]_{6}$
181.6.5.1$x^{6} - 181$$6$$1$$5$$C_6$$[\ ]_{6}$