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Properties

Label 18.18.3340378829...1232.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{24}\cdot 3^{18}\cdot 7^{12}\cdot 13^{5}$
Root discriminant $56.41$
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![104, -312, -2808, 11648, -5070, -27612, 26549, 24654, -32508, -10182, 18303, 1908, -5266, -114, 756, -4, -48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 48*x^16 - 4*x^15 + 756*x^14 - 114*x^13 - 5266*x^12 + 1908*x^11 + 18303*x^10 - 10182*x^9 - 32508*x^8 + 24654*x^7 + 26549*x^6 - 27612*x^5 - 5070*x^4 + 11648*x^3 - 2808*x^2 - 312*x + 104)
 
gp: K = bnfinit(x^18 - 48*x^16 - 4*x^15 + 756*x^14 - 114*x^13 - 5266*x^12 + 1908*x^11 + 18303*x^10 - 10182*x^9 - 32508*x^8 + 24654*x^7 + 26549*x^6 - 27612*x^5 - 5070*x^4 + 11648*x^3 - 2808*x^2 - 312*x + 104, 1)
 

Normalized defining polynomial

\( x^{18} - 48 x^{16} - 4 x^{15} + 756 x^{14} - 114 x^{13} - 5266 x^{12} + 1908 x^{11} + 18303 x^{10} - 10182 x^{9} - 32508 x^{8} + 24654 x^{7} + 26549 x^{6} - 27612 x^{5} - 5070 x^{4} + 11648 x^{3} - 2808 x^{2} - 312 x + 104 \)

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:  \(33403788296933455481250074591232=2^{24}\cdot 3^{18}\cdot 7^{12}\cdot 13^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{6} a^{14} - \frac{1}{3} a^{8} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{6} a^{2}$, $\frac{1}{126} a^{15} + \frac{1}{14} a^{14} + \frac{1}{21} a^{13} + \frac{8}{63} a^{12} + \frac{2}{7} a^{11} + \frac{4}{21} a^{10} - \frac{16}{63} a^{9} + \frac{3}{7} a^{8} - \frac{3}{14} a^{7} - \frac{7}{18} a^{6} - \frac{1}{7} a^{5} + \frac{2}{21} a^{4} - \frac{5}{14} a^{3} + \frac{3}{14} a^{2} + \frac{5}{21} a + \frac{10}{63}$, $\frac{1}{252} a^{16} - \frac{1}{21} a^{14} + \frac{1}{63} a^{13} - \frac{2}{21} a^{12} + \frac{13}{42} a^{11} - \frac{61}{126} a^{10} - \frac{1}{7} a^{9} - \frac{1}{28} a^{8} + \frac{55}{126} a^{7} - \frac{5}{21} a^{6} - \frac{13}{42} a^{5} - \frac{37}{84} a^{4} + \frac{1}{21} a^{3} + \frac{17}{42} a^{2} - \frac{10}{63} a - \frac{1}{21}$, $\frac{1}{96511385813691492} a^{17} + \frac{108795635268599}{96511385813691492} a^{16} + \frac{8627169736624}{8042615484474291} a^{15} + \frac{3789277576937513}{48255692906845746} a^{14} + \frac{787864044546722}{24127846453422873} a^{13} + \frac{436447254278569}{5361743656316194} a^{12} - \frac{4968771699379037}{24127846453422873} a^{11} + \frac{9941913882081283}{48255692906845746} a^{10} + \frac{7317611879463701}{32170461937897164} a^{9} + \frac{27702182240682743}{96511385813691492} a^{8} + \frac{3248480879753543}{6893670415263678} a^{7} - \frac{3027059590579505}{8042615484474291} a^{6} - \frac{7105073322100343}{32170461937897164} a^{5} - \frac{1972553770129997}{4595780276842452} a^{4} - \frac{30639429118837}{765963379473742} a^{3} - \frac{4956761293590289}{24127846453422873} a^{2} + \frac{54199348278622}{24127846453422873} a + \frac{2792564324912651}{8042615484474291}$

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:  \( 11075540258.1 \) (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

\(\Q(\zeta_{7})^+\), 6.6.1997632.1, 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R $18$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $18$

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
$2$2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.12.18.57$x^{12} + 14 x^{11} + 16 x^{10} + 6 x^{8} + 4 x^{7} - 4 x^{6} + 8 x^{2} + 16 x - 8$$4$$3$$18$$C_2^2 \times A_4$$[2, 2, 2]^{6}$
$3$3.9.9.5$x^{9} + 3 x^{7} + 3 x^{6} + 54$$3$$3$$9$$(C_3^2:C_3):C_2$$[3/2, 3/2, 3/2]_{2}^{3}$
3.9.9.5$x^{9} + 3 x^{7} + 3 x^{6} + 54$$3$$3$$9$$(C_3^2:C_3):C_2$$[3/2, 3/2, 3/2]_{2}^{3}$
$7$7.6.4.3$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}$
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}$