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Properties

Label 18.6.12137503248...2992.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 37^{9}\cdot 151^{2}$
Root discriminant $16.86$
Ramified primes $2, 37, 151$
Class number $1$
Class group Trivial
Galois group 18T319

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -40, -12, 126, -144, 109, -71, 82, -16, -38, 31, -26, 33, -40, 20, -5, 7, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 7*x^16 - 5*x^15 + 20*x^14 - 40*x^13 + 33*x^12 - 26*x^11 + 31*x^10 - 38*x^9 - 16*x^8 + 82*x^7 - 71*x^6 + 109*x^5 - 144*x^4 + 126*x^3 - 12*x^2 - 40*x + 25)
 
gp: K = bnfinit(x^18 - 5*x^17 + 7*x^16 - 5*x^15 + 20*x^14 - 40*x^13 + 33*x^12 - 26*x^11 + 31*x^10 - 38*x^9 - 16*x^8 + 82*x^7 - 71*x^6 + 109*x^5 - 144*x^4 + 126*x^3 - 12*x^2 - 40*x + 25, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 7 x^{16} - 5 x^{15} + 20 x^{14} - 40 x^{13} + 33 x^{12} - 26 x^{11} + 31 x^{10} - 38 x^{9} - 16 x^{8} + 82 x^{7} - 71 x^{6} + 109 x^{5} - 144 x^{4} + 126 x^{3} - 12 x^{2} - 40 x + 25 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12137503248660687572992=2^{12}\cdot 37^{9}\cdot 151^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.86$
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:  $2, 37, 151$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{12} - \frac{2}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{12} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{12} - \frac{2}{5} a^{11} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a$, $\frac{1}{11488619764285} a^{17} + \frac{347778536552}{11488619764285} a^{16} + \frac{500730392673}{11488619764285} a^{15} - \frac{50623880311}{2297723952857} a^{14} + \frac{472130680154}{11488619764285} a^{13} + \frac{4141726685507}{11488619764285} a^{12} + \frac{3350186629266}{11488619764285} a^{11} - \frac{271346128718}{2297723952857} a^{10} + \frac{2277264800267}{11488619764285} a^{9} - \frac{5226512201691}{11488619764285} a^{8} + \frac{161443406034}{11488619764285} a^{7} + \frac{1495308455787}{11488619764285} a^{6} - \frac{958230896674}{2297723952857} a^{5} + \frac{546660597567}{11488619764285} a^{4} + \frac{244919490642}{11488619764285} a^{3} - \frac{1869038819522}{11488619764285} a^{2} - \frac{3983234298671}{11488619764285} a + \frac{846122096175}{2297723952857}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 10280.4093864 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T319:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1296
The 22 conjugacy class representatives for t18n319
Character table for t18n319 is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.148.1 x3, 6.6.810448.1, 9.3.489510592.1

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
Degree 12 sibling: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
37Data not computed
$151$$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$