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Properties

Label 18.0.47963521726...4144.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 7^{15}\cdot 11^{9}$
Root discriminant $96.00$
Ramified primes $2, 3, 7, 11$
Class number $130248$ (GRH)
Class group $[6, 6, 3618]$ (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![7031654056, 2047765272, 6255389784, 932445416, 749479584, -24671388, 140163682, -61654620, 25552602, -9052937, 2543787, -670416, 164170, -31242, 6498, -828, 132, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 132*x^16 - 828*x^15 + 6498*x^14 - 31242*x^13 + 164170*x^12 - 670416*x^11 + 2543787*x^10 - 9052937*x^9 + 25552602*x^8 - 61654620*x^7 + 140163682*x^6 - 24671388*x^5 + 749479584*x^4 + 932445416*x^3 + 6255389784*x^2 + 2047765272*x + 7031654056)
 
gp: K = bnfinit(x^18 - 9*x^17 + 132*x^16 - 828*x^15 + 6498*x^14 - 31242*x^13 + 164170*x^12 - 670416*x^11 + 2543787*x^10 - 9052937*x^9 + 25552602*x^8 - 61654620*x^7 + 140163682*x^6 - 24671388*x^5 + 749479584*x^4 + 932445416*x^3 + 6255389784*x^2 + 2047765272*x + 7031654056, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 132 x^{16} - 828 x^{15} + 6498 x^{14} - 31242 x^{13} + 164170 x^{12} - 670416 x^{11} + 2543787 x^{10} - 9052937 x^{9} + 25552602 x^{8} - 61654620 x^{7} + 140163682 x^{6} - 24671388 x^{5} + 749479584 x^{4} + 932445416 x^{3} + 6255389784 x^{2} + 2047765272 x + 7031654056 \)

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:  \(-479635217269575679584198692398854144=-\,2^{12}\cdot 3^{21}\cdot 7^{15}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.00$
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:  $2, 3, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{6}$, $\frac{1}{28} a^{15} - \frac{1}{14} a^{14} - \frac{1}{28} a^{13} - \frac{1}{14} a^{12} - \frac{5}{28} a^{11} + \frac{3}{14} a^{10} - \frac{1}{28} a^{9} + \frac{1}{14} a^{8} - \frac{3}{14} a^{7} + \frac{1}{14} a^{6} + \frac{3}{7} a^{5} - \frac{3}{14} a^{4} + \frac{2}{7} a^{3}$, $\frac{1}{140} a^{16} - \frac{3}{35} a^{14} - \frac{11}{140} a^{13} + \frac{3}{35} a^{12} - \frac{5}{28} a^{11} + \frac{8}{35} a^{10} + \frac{1}{20} a^{9} - \frac{23}{140} a^{8} - \frac{17}{140} a^{7} + \frac{4}{35} a^{6} + \frac{8}{35} a^{5} + \frac{33}{70} a^{4} + \frac{29}{70} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{46117212652671013857083701943962630993798415658942799656141818533611010741700} a^{17} - \frac{14309228415199943974225766780142565985336136963540198588353284343614546188}{11529303163167753464270925485990657748449603914735699914035454633402752685425} a^{16} - \frac{655413108304762075011249348282473479692700319790489767254658634806108166107}{46117212652671013857083701943962630993798415658942799656141818533611010741700} a^{15} + \frac{983428740065403573894538353632213391793545921544187411225005941249347175162}{11529303163167753464270925485990657748449603914735699914035454633402752685425} a^{14} + \frac{206569420507790832958993488626040885725048445619538065550424233567661823287}{6588173236095859122440528849137518713399773665563257093734545504801572963100} a^{13} - \frac{93437794739319629292309342596272908368651580912622194182812299978835867076}{11529303163167753464270925485990657748449603914735699914035454633402752685425} a^{12} + \frac{567306314769731581593004842135012193160839793804158091719181876922181988017}{46117212652671013857083701943962630993798415658942799656141818533611010741700} a^{11} - \frac{315566839413421100710471590362025793448024373653249134546017289540128521049}{1647043309023964780610132212284379678349943416390814273433636376200393240775} a^{10} + \frac{2144774717032443492528168386160112445637125546878092653817341530143913193329}{23058606326335506928541850971981315496899207829471399828070909266805505370850} a^{9} - \frac{3543618944156730267525418964755662025786081664476596858545170942364033610203}{23058606326335506928541850971981315496899207829471399828070909266805505370850} a^{8} - \frac{17868595827187369314882755936000268916836197814813014292541007409928380677}{658817323609585912244052884913751871339977366556325709373454550480157296310} a^{7} + \frac{3368394744981939286024647014219297338189922714088551829917051738072574799}{35474778963593087582372078418432793072152627429955999735493706564316162109} a^{6} - \frac{3297752960366563749712041308554479625466812712978589837181198670926916047517}{11529303163167753464270925485990657748449603914735699914035454633402752685425} a^{5} - \frac{2846573932739739941016782339502368554360873377935669996395172834801249765907}{23058606326335506928541850971981315496899207829471399828070909266805505370850} a^{4} + \frac{4270563074250256104953035551044629178822183977197367910985287606405166348034}{11529303163167753464270925485990657748449603914735699914035454633402752685425} a^{3} + \frac{8581924003128594163014073500602866988318129649047974394080985753410058237}{126695639155689598508471708637259975257687955106985713341048952015414864675} a^{2} - \frac{70924443803941455632751448842387025504880823681061281738985108567125020606}{329408661804792956122026442456875935669988683278162854686727275240078648155} a - \frac{104929590418997526759246764212202492718663584863433221277940708036867599986}{1647043309023964780610132212284379678349943416390814273433636376200393240775}$

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

Class group and class number

$C_{6}\times C_{6}\times C_{3618}$, which has order $130248$ (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:  \( 296124.35954857944 \) (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

\(\Q(\sqrt{-231}) \), 3.3.756.1, \(\Q(\zeta_{7})^+\), 6.0.15975002736.2, 6.0.603993159.1, 9.9.1037426999616.1

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$