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Properties

Label 18.6.41619125031...1857.2
Degree $18$
Signature $[6, 6]$
Discriminant $19^{16}\cdot 113^{3}$
Root discriminant $30.12$
Ramified primes $19, 113$
Class number $1$
Class group Trivial
Galois group 18T264

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -151, 375, 13, 312, 161, -87, -45, -819, 1210, -708, -58, 524, -507, 292, -120, 35, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 35*x^16 - 120*x^15 + 292*x^14 - 507*x^13 + 524*x^12 - 58*x^11 - 708*x^10 + 1210*x^9 - 819*x^8 - 45*x^7 - 87*x^6 + 161*x^5 + 312*x^4 + 13*x^3 + 375*x^2 - 151*x - 1)
 
gp: K = bnfinit(x^18 - 8*x^17 + 35*x^16 - 120*x^15 + 292*x^14 - 507*x^13 + 524*x^12 - 58*x^11 - 708*x^10 + 1210*x^9 - 819*x^8 - 45*x^7 - 87*x^6 + 161*x^5 + 312*x^4 + 13*x^3 + 375*x^2 - 151*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} + 35 x^{16} - 120 x^{15} + 292 x^{14} - 507 x^{13} + 524 x^{12} - 58 x^{11} - 708 x^{10} + 1210 x^{9} - 819 x^{8} - 45 x^{7} - 87 x^{6} + 161 x^{5} + 312 x^{4} + 13 x^{3} + 375 x^{2} - 151 x - 1 \)

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:  \(416191250312479879983411857=19^{16}\cdot 113^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.12$
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:  $19, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{3}{7} a^{14} - \frac{2}{7} a^{13} - \frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{3}{7} a^{10} + \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{16} + \frac{3}{7} a^{14} - \frac{2}{7} a^{13} - \frac{1}{7} a^{11} + \frac{2}{7} a^{10} + \frac{2}{7} a^{9} - \frac{2}{7} a^{8} + \frac{3}{7} a^{7} + \frac{3}{7} a^{6} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} + \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{978539116614232700668423429} a^{17} + \frac{50520993873484958451081952}{978539116614232700668423429} a^{16} + \frac{12280987868442265148482310}{978539116614232700668423429} a^{15} - \frac{75233247378879719445192922}{978539116614232700668423429} a^{14} - \frac{314498256112904451610964176}{978539116614232700668423429} a^{13} - \frac{293941207335865207584481703}{978539116614232700668423429} a^{12} - \frac{231989966505751991569555497}{978539116614232700668423429} a^{11} + \frac{466834238788378495123783051}{978539116614232700668423429} a^{10} - \frac{61362518086073304675439080}{139791302373461814381203347} a^{9} + \frac{139406548516608654436054714}{978539116614232700668423429} a^{8} - \frac{420314209207270287748058168}{978539116614232700668423429} a^{7} - \frac{246147925990073226982771247}{978539116614232700668423429} a^{6} + \frac{290282028576413601428675262}{978539116614232700668423429} a^{5} + \frac{140944696309131446131734752}{978539116614232700668423429} a^{4} + \frac{127273681132909110289417178}{978539116614232700668423429} a^{3} + \frac{430692785531718247844270827}{978539116614232700668423429} a^{2} + \frac{237730554188672474178060914}{978539116614232700668423429} a + \frac{280247996675464329145465928}{978539116614232700668423429}$

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:  \( 1779210.10134 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T264:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1152
The 32 conjugacy class representatives for t18n264
Character table for t18n264 is not computed

Intermediate fields

3.3.361.1, \(\Q(\zeta_{19})^+\)

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

Sibling fields

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ R $18$ $18$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
19Data not computed
$113$$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$