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Properties

Label 18.18.1446519374...3152.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{16}\cdot 19^{8}\cdot 37^{9}$
Root discriminant $41.69$
Ramified primes $2, 19, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_9:C_3$ (as 18T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-137, 865, 1755, -16692, 18518, 29002, -49858, -13752, 45321, -2623, -19161, 4144, 3998, -1330, -362, 180, 3, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 3*x^16 + 180*x^15 - 362*x^14 - 1330*x^13 + 3998*x^12 + 4144*x^11 - 19161*x^10 - 2623*x^9 + 45321*x^8 - 13752*x^7 - 49858*x^6 + 29002*x^5 + 18518*x^4 - 16692*x^3 + 1755*x^2 + 865*x - 137)
 
gp: K = bnfinit(x^18 - 9*x^17 + 3*x^16 + 180*x^15 - 362*x^14 - 1330*x^13 + 3998*x^12 + 4144*x^11 - 19161*x^10 - 2623*x^9 + 45321*x^8 - 13752*x^7 - 49858*x^6 + 29002*x^5 + 18518*x^4 - 16692*x^3 + 1755*x^2 + 865*x - 137, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 3 x^{16} + 180 x^{15} - 362 x^{14} - 1330 x^{13} + 3998 x^{12} + 4144 x^{11} - 19161 x^{10} - 2623 x^{9} + 45321 x^{8} - 13752 x^{7} - 49858 x^{6} + 29002 x^{5} + 18518 x^{4} - 16692 x^{3} + 1755 x^{2} + 865 x - 137 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(144651937430092424868603953152=2^{16}\cdot 19^{8}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.69$
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:  $2, 19, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{22044833341} a^{16} - \frac{8}{22044833341} a^{15} + \frac{8914960863}{22044833341} a^{14} + \frac{3729774122}{22044833341} a^{13} + \frac{7348244352}{22044833341} a^{12} - \frac{4397196698}{22044833341} a^{11} + \frac{72917372}{22044833341} a^{10} - \frac{10781277427}{22044833341} a^{9} - \frac{3207471477}{22044833341} a^{8} - \frac{418314723}{22044833341} a^{7} + \frac{4638546891}{22044833341} a^{6} - \frac{1465779057}{22044833341} a^{5} - \frac{3972333798}{22044833341} a^{4} - \frac{3725562789}{22044833341} a^{3} + \frac{8469504587}{22044833341} a^{2} - \frac{5206012211}{22044833341} a + \frac{7843549898}{22044833341}$, $\frac{1}{8090453836147} a^{17} + \frac{175}{8090453836147} a^{16} - \frac{1313775041061}{8090453836147} a^{15} + \frac{3398754279331}{8090453836147} a^{14} - \frac{698927591805}{8090453836147} a^{13} + \frac{458538186078}{8090453836147} a^{12} + \frac{3097321422995}{8090453836147} a^{11} + \frac{3595870436232}{8090453836147} a^{10} - \frac{2439122740779}{8090453836147} a^{9} - \frac{256713094899}{8090453836147} a^{8} - \frac{689168380966}{8090453836147} a^{7} + \frac{3580947636280}{8090453836147} a^{6} - \frac{2675096735398}{8090453836147} a^{5} + \frac{834520519388}{8090453836147} a^{4} - \frac{2436895153080}{8090453836147} a^{3} - \frac{1497473673848}{8090453836147} a^{2} + \frac{2648451149868}{8090453836147} a - \frac{1364324202973}{8090453836147}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:  \( 698998807.188 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9:C_3$ (as 18T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $D_9:C_3$
Character table for $D_9:C_3$

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.148.1 x3, 6.6.810448.1, 9.9.62526089134336.1 x3

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

Sibling fields

Degree 9 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 ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
2Data not computed
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.6.4.3$x^{6} + 95 x^{3} + 2888$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.4.3$x^{6} + 95 x^{3} + 2888$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$37$37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$