↑

Properties

Label 18.0.48432172600...8272.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{14}\cdot 47^{11}$
Root discriminant $39.23$
Ramified primes $2, 3, 47$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3:S_3:S_4$ (as 18T153)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1181, -1375, 3492, -3340, 6151, -7007, 10396, -10565, 11270, -8613, 6692, -3775, 2210, -907, 403, -112, 36, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 36*x^16 - 112*x^15 + 403*x^14 - 907*x^13 + 2210*x^12 - 3775*x^11 + 6692*x^10 - 8613*x^9 + 11270*x^8 - 10565*x^7 + 10396*x^6 - 7007*x^5 + 6151*x^4 - 3340*x^3 + 3492*x^2 - 1375*x + 1181)
 
gp: K = bnfinit(x^18 - 5*x^17 + 36*x^16 - 112*x^15 + 403*x^14 - 907*x^13 + 2210*x^12 - 3775*x^11 + 6692*x^10 - 8613*x^9 + 11270*x^8 - 10565*x^7 + 10396*x^6 - 7007*x^5 + 6151*x^4 - 3340*x^3 + 3492*x^2 - 1375*x + 1181, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 36 x^{16} - 112 x^{15} + 403 x^{14} - 907 x^{13} + 2210 x^{12} - 3775 x^{11} + 6692 x^{10} - 8613 x^{9} + 11270 x^{8} - 10565 x^{7} + 10396 x^{6} - 7007 x^{5} + 6151 x^{4} - 3340 x^{3} + 3492 x^{2} - 1375 x + 1181 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-48432172600570524634593718272=-\,2^{12}\cdot 3^{14}\cdot 47^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.23$
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:  $2, 3, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} - \frac{1}{7} a^{15} - \frac{3}{7} a^{13} - \frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{3}{7} a^{10} - \frac{2}{7} a^{9} - \frac{3}{7} a^{8} - \frac{2}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{4} + \frac{3}{7} a^{2} - \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{272600759284224003512573533} a^{17} + \frac{1904538258485119257905517}{272600759284224003512573533} a^{16} + \frac{42122387392505213043171259}{272600759284224003512573533} a^{15} + \frac{66323106718858616519134022}{272600759284224003512573533} a^{14} - \frac{43298923398450675593224792}{272600759284224003512573533} a^{13} - \frac{77233794175415882987410800}{272600759284224003512573533} a^{12} + \frac{75433504023885683670966009}{272600759284224003512573533} a^{11} + \frac{120983471959659005008132783}{272600759284224003512573533} a^{10} + \frac{78316757389705421723070908}{272600759284224003512573533} a^{9} + \frac{58600370813024942836618417}{272600759284224003512573533} a^{8} + \frac{124837041586071200141955988}{272600759284224003512573533} a^{7} - \frac{58238163474053771504364410}{272600759284224003512573533} a^{6} - \frac{6515557772156497127007935}{272600759284224003512573533} a^{5} - \frac{46559608743577169590350029}{272600759284224003512573533} a^{4} + \frac{51090505780653525489412952}{272600759284224003512573533} a^{3} + \frac{87905265419618689228118364}{272600759284224003512573533} a^{2} + \frac{60326622018864394278068853}{272600759284224003512573533} a - \frac{82490357475171091389425551}{272600759284224003512573533}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $8$
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:  \( 3723839.32807 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3:S_4$ (as 18T153):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 20 conjugacy class representatives for $C_3:S_3:S_4$
Character table for $C_3:S_3:S_4$

Intermediate fields

3.3.564.1, 6.0.14950512.2, 9.3.682999190208.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.12.14.15$x^{12} + 9 x^{11} - 6 x^{10} + 6 x^{9} - 3 x^{8} + 9 x^{7} + 6 x^{6} - 9 x^{5} - 9 x^{4} - 9 x^{3} - 9 x^{2} + 9$$6$$2$$14$$C_6\times S_3$$[3/2]_{2}^{6}$
47Data not computed