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Properties

Label 18.0.32649949405...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 5^{9}\cdot 151^{9}$
Root discriminant $43.62$
Ramified primes $2, 5, 151$
Class number $36$ (GRH)
Class group $[3, 12]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![471875, 0, 564716, 0, 291260, 0, 36851, 0, 32200, 0, 5776, 0, 1645, 0, 156, 0, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 20*x^16 + 156*x^14 + 1645*x^12 + 5776*x^10 + 32200*x^8 + 36851*x^6 + 291260*x^4 + 564716*x^2 + 471875)
 
gp: K = bnfinit(x^18 + 20*x^16 + 156*x^14 + 1645*x^12 + 5776*x^10 + 32200*x^8 + 36851*x^6 + 291260*x^4 + 564716*x^2 + 471875, 1)
 

Normalized defining polynomial

\( x^{18} + 20 x^{16} + 156 x^{14} + 1645 x^{12} + 5776 x^{10} + 32200 x^{8} + 36851 x^{6} + 291260 x^{4} + 564716 x^{2} + 471875 \)

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:  \(-326499494057574490808000000000=-\,2^{12}\cdot 5^{9}\cdot 151^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.62$
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:  $2, 5, 151$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{10} a^{9} - \frac{1}{2} a^{6} - \frac{1}{5} a^{5} - \frac{1}{2} a^{3} - \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{10} a^{10} - \frac{1}{2} a^{7} - \frac{1}{5} a^{6} - \frac{1}{2} a^{4} - \frac{2}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{11} - \frac{1}{2} a^{8} - \frac{1}{5} a^{7} - \frac{1}{2} a^{5} - \frac{2}{5} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2220} a^{12} + \frac{22}{555} a^{10} + \frac{32}{555} a^{8} + \frac{67}{185} a^{6} - \frac{211}{555} a^{4} - \frac{1}{2} a^{3} + \frac{67}{555} a^{2} - \frac{187}{444}$, $\frac{1}{11100} a^{13} + \frac{31}{1110} a^{11} - \frac{269}{5550} a^{9} - \frac{1}{2} a^{8} + \frac{6}{185} a^{7} - \frac{1}{2} a^{6} + \frac{799}{5550} a^{5} - \frac{1}{2} a^{4} + \frac{493}{1110} a^{3} - \frac{1}{2} a^{2} - \frac{2711}{11100} a - \frac{1}{2}$, $\frac{1}{444166500} a^{14} + \frac{5329}{88833300} a^{12} + \frac{180221}{37013875} a^{10} + \frac{10392772}{22208325} a^{8} + \frac{17057792}{111041625} a^{6} - \frac{1}{2} a^{5} - \frac{2240251}{7402775} a^{4} - \frac{1}{2} a^{3} + \frac{22956929}{444166500} a^{2} + \frac{203}{23532}$, $\frac{1}{444166500} a^{15} - \frac{1337}{44416650} a^{13} - \frac{5120999}{222083250} a^{11} + \frac{121671}{7402775} a^{9} - \frac{1}{2} a^{8} + \frac{13456442}{111041625} a^{7} - \frac{1}{2} a^{6} - \frac{19835903}{44416650} a^{5} + \frac{47766229}{444166500} a^{3} - \frac{1}{2} a^{2} + \frac{24793}{98050} a$, $\frac{1}{8047852813500} a^{16} - \frac{1}{80478528135} a^{14} - \frac{1560776623}{8047852813500} a^{12} - \frac{1093636643}{80478528135} a^{10} - \frac{198086392761}{670654401125} a^{8} - \frac{1}{2} a^{7} - \frac{2028041787}{134130880225} a^{6} - \frac{835111004457}{2682617604500} a^{4} - \frac{1}{2} a^{3} + \frac{48825884867}{134130880225} a^{2} + \frac{40010527}{426376308}$, $\frac{1}{8047852813500} a^{17} - \frac{1}{80478528135} a^{15} - \frac{36904351}{2682617604500} a^{13} + \frac{3401560424}{80478528135} a^{11} + \frac{418717646}{54377383875} a^{9} - \frac{1}{2} a^{8} + \frac{6672339633}{134130880225} a^{7} + \frac{38417273897}{217509535500} a^{5} - \frac{1}{2} a^{4} + \frac{101525683931}{402392640675} a^{3} + \frac{515691}{96030700} a$

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

Class group and class number

$C_{3}\times C_{12}$, which has order $36$ (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:  \( 7146431.9696 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-755}) \), 3.1.3020.2 x3, 3.1.3020.3 x3, 3.1.3020.1 x3, 3.1.755.1 x3, 6.0.6885902000.2, 6.0.6885902000.3, 6.0.6885902000.1, 6.0.430368875.1, 9.1.20795424040000.2 x9

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$151$151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$