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Properties

Label 18.0.92176615928...6571.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 7^{12}\cdot 11^{9}$
Root discriminant $52.51$
Ramified primes $3, 7, 11$
Class number $1296$ (GRH)
Class group $[36, 36]$ (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![3180563, -3981963, 5618562, -4490689, 3606933, -2079090, 1183880, -534324, 242916, -94398, 36576, -11892, 3178, -540, 84, -52, 33, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 33*x^16 - 52*x^15 + 84*x^14 - 540*x^13 + 3178*x^12 - 11892*x^11 + 36576*x^10 - 94398*x^9 + 242916*x^8 - 534324*x^7 + 1183880*x^6 - 2079090*x^5 + 3606933*x^4 - 4490689*x^3 + 5618562*x^2 - 3981963*x + 3180563)
 
gp: K = bnfinit(x^18 - 9*x^17 + 33*x^16 - 52*x^15 + 84*x^14 - 540*x^13 + 3178*x^12 - 11892*x^11 + 36576*x^10 - 94398*x^9 + 242916*x^8 - 534324*x^7 + 1183880*x^6 - 2079090*x^5 + 3606933*x^4 - 4490689*x^3 + 5618562*x^2 - 3981963*x + 3180563, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 33 x^{16} - 52 x^{15} + 84 x^{14} - 540 x^{13} + 3178 x^{12} - 11892 x^{11} + 36576 x^{10} - 94398 x^{9} + 242916 x^{8} - 534324 x^{7} + 1183880 x^{6} - 2079090 x^{5} + 3606933 x^{4} - 4490689 x^{3} + 5618562 x^{2} - 3981963 x + 3180563 \)

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:  \(-9217661592820801741280239766571=-\,3^{24}\cdot 7^{12}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.51$
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:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(693=3^{2}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{693}(1,·)$, $\chi_{693}(67,·)$, $\chi_{693}(331,·)$, $\chi_{693}(142,·)$, $\chi_{693}(463,·)$, $\chi_{693}(529,·)$, $\chi_{693}(274,·)$, $\chi_{693}(340,·)$, $\chi_{693}(604,·)$, $\chi_{693}(100,·)$, $\chi_{693}(232,·)$, $\chi_{693}(298,·)$, $\chi_{693}(43,·)$, $\chi_{693}(109,·)$, $\chi_{693}(562,·)$, $\chi_{693}(373,·)$, $\chi_{693}(505,·)$, $\chi_{693}(571,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{36068} a^{16} - \frac{3745}{18034} a^{15} + \frac{2063}{18034} a^{14} + \frac{3701}{18034} a^{13} + \frac{1819}{18034} a^{12} - \frac{31}{18034} a^{11} + \frac{1608}{9017} a^{10} - \frac{1285}{9017} a^{9} - \frac{1409}{18034} a^{8} - \frac{4453}{9017} a^{7} + \frac{858}{9017} a^{6} - \frac{56}{127} a^{5} - \frac{953}{9017} a^{4} + \frac{1881}{18034} a^{3} + \frac{11255}{36068} a^{2} + \frac{4277}{18034} a - \frac{17965}{36068}$, $\frac{1}{21776211946736141773776215993848042042394812} a^{17} - \frac{65276629670209803952168366735669733339}{10888105973368070886888107996924021021197406} a^{16} - \frac{1898431225563290775084581979915384771334967}{10888105973368070886888107996924021021197406} a^{15} - \frac{194110183337236842419447889737822921435005}{5444052986684035443444053998462010510598703} a^{14} - \frac{1109311798914616209080761794087776527165388}{5444052986684035443444053998462010510598703} a^{13} - \frac{2399071067892754821476347022774272154519351}{10888105973368070886888107996924021021197406} a^{12} + \frac{1302962743920113870900147324961573271844053}{5444052986684035443444053998462010510598703} a^{11} + \frac{232922639325320632172161762973574618320621}{10888105973368070886888107996924021021197406} a^{10} + \frac{1074483305771666395418510320435087702061129}{5444052986684035443444053998462010510598703} a^{9} - \frac{1176536509540153208877405710006959118478764}{5444052986684035443444053998462010510598703} a^{8} - \frac{958955811957143233841962095167990006062023}{10888105973368070886888107996924021021197406} a^{7} + \frac{1242470345265288794665371874553799074938111}{5444052986684035443444053998462010510598703} a^{6} - \frac{4209479181815331690645318865696016954667889}{10888105973368070886888107996924021021197406} a^{5} - \frac{989949539637290109526089555824942917936715}{10888105973368070886888107996924021021197406} a^{4} + \frac{1398023879586111942252789796958976338404057}{21776211946736141773776215993848042042394812} a^{3} + \frac{2243752140223369634358852438424913943429595}{10888105973368070886888107996924021021197406} a^{2} - \frac{5846766751963184980943904744232230958010607}{21776211946736141773776215993848042042394812} a - \frac{147246568373702776579226282106419720596184}{5444052986684035443444053998462010510598703}$

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

Class group and class number

$C_{36}\times C_{36}$, which has order $1296$ (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:  \( 54408.4888887 \) (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

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.1, \(\Q(\zeta_{7})^+\), 3.3.3969.2, 6.0.8732691.1, 6.0.20967191091.4, 6.0.3195731.1, 6.0.20967191091.5, 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$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
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$