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Properties

Label 18.18.9019022840...3264.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{24}\cdot 3^{21}\cdot 7^{12}\cdot 13^{5}$
Root discriminant $67.74$
Ramified primes $2, 3, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T282

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-351, 0, 7020, 0, -34047, 0, 70731, 0, -71541, 0, 35514, 0, -8442, 0, 963, 0, -51, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 51*x^16 + 963*x^14 - 8442*x^12 + 35514*x^10 - 71541*x^8 + 70731*x^6 - 34047*x^4 + 7020*x^2 - 351)
 
gp: K = bnfinit(x^18 - 51*x^16 + 963*x^14 - 8442*x^12 + 35514*x^10 - 71541*x^8 + 70731*x^6 - 34047*x^4 + 7020*x^2 - 351, 1)
 

Normalized defining polynomial

\( x^{18} - 51 x^{16} + 963 x^{14} - 8442 x^{12} + 35514 x^{10} - 71541 x^{8} + 70731 x^{6} - 34047 x^{4} + 7020 x^{2} - 351 \)

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:  \(901902284017203297993752013963264=2^{24}\cdot 3^{21}\cdot 7^{12}\cdot 13^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.74$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{6} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{6} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{36} a^{12} - \frac{1}{12} a^{8} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{36} a^{13} - \frac{1}{12} a^{9} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{36} a^{14} - \frac{1}{12} a^{10} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{72} a^{15} - \frac{1}{72} a^{14} - \frac{1}{72} a^{12} - \frac{1}{24} a^{11} + \frac{1}{24} a^{10} + \frac{1}{24} a^{8} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{1}{8}$, $\frac{1}{105595992} a^{16} - \frac{775037}{105595992} a^{14} - \frac{1}{72} a^{13} - \frac{188275}{52797996} a^{12} + \frac{1659785}{35198664} a^{10} + \frac{1}{24} a^{9} - \frac{2815901}{35198664} a^{8} - \frac{1270315}{17599332} a^{6} - \frac{1}{2} a^{5} + \frac{1323827}{11732888} a^{4} + \frac{1}{4} a^{3} - \frac{3135931}{11732888} a^{2} + \frac{3}{8} a + \frac{600589}{11732888}$, $\frac{1}{105595992} a^{17} + \frac{345787}{52797996} a^{15} - \frac{188275}{52797996} a^{13} - \frac{1}{72} a^{12} + \frac{96587}{17599332} a^{11} - \frac{2815901}{35198664} a^{9} + \frac{1}{24} a^{8} - \frac{1270315}{17599332} a^{7} + \frac{4257049}{11732888} a^{5} - \frac{1}{2} a^{4} + \frac{1049281}{2933222} a^{3} + \frac{1}{4} a^{2} - \frac{5265855}{11732888} a + \frac{3}{8}$

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:  \( 79934267357.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T282:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1296
The 56 conjugacy class representatives for t18n282 are not computed
Character table for t18n282 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.6.53936064.1, 9.9.25046451847872.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R $18$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ $18$

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
$2$2.6.6.4$x^{6} + x^{2} + 1$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.12.18.57$x^{12} + 14 x^{11} + 16 x^{10} + 6 x^{8} + 4 x^{7} - 4 x^{6} + 8 x^{2} + 16 x - 8$$4$$3$$18$$C_2^2 \times A_4$$[2, 2, 2]^{6}$
3Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.5.6$x^{6} + 416$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$