↑

Properties

Label 18.0.91843363121...2928.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 11^{8}$
Root discriminant $16.60$
Ramified primes $2, 3, 11$
Class number $1$
Class group Trivial
Galois group $C_3^2:S_3$ (as 18T24)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -120, 357, -770, 1371, -2088, 2810, -3294, 3360, -2970, 2301, -1560, 927, -474, 207, -76, 24, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 24*x^16 - 76*x^15 + 207*x^14 - 474*x^13 + 927*x^12 - 1560*x^11 + 2301*x^10 - 2970*x^9 + 3360*x^8 - 3294*x^7 + 2810*x^6 - 2088*x^5 + 1371*x^4 - 770*x^3 + 357*x^2 - 120*x + 25)
 
gp: K = bnfinit(x^18 - 6*x^17 + 24*x^16 - 76*x^15 + 207*x^14 - 474*x^13 + 927*x^12 - 1560*x^11 + 2301*x^10 - 2970*x^9 + 3360*x^8 - 3294*x^7 + 2810*x^6 - 2088*x^5 + 1371*x^4 - 770*x^3 + 357*x^2 - 120*x + 25, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 24 x^{16} - 76 x^{15} + 207 x^{14} - 474 x^{13} + 927 x^{12} - 1560 x^{11} + 2301 x^{10} - 2970 x^{9} + 3360 x^{8} - 3294 x^{7} + 2810 x^{6} - 2088 x^{5} + 1371 x^{4} - 770 x^{3} + 357 x^{2} - 120 x + 25 \)

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:  \(-9184336312155528572928=-\,2^{12}\cdot 3^{21}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.60$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{11} - \frac{1}{10} a^{10} + \frac{1}{10} a^{9} - \frac{1}{2} a^{6} - \frac{1}{5} a^{5} + \frac{3}{10} a^{4} + \frac{2}{5} a^{3} - \frac{1}{10} a$, $\frac{1}{10} a^{12} + \frac{1}{10} a^{9} - \frac{1}{2} a^{7} + \frac{3}{10} a^{6} + \frac{1}{10} a^{5} - \frac{3}{10} a^{4} + \frac{2}{5} a^{3} - \frac{1}{10} a^{2} - \frac{1}{10} a$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{10} - \frac{1}{2} a^{8} + \frac{3}{10} a^{7} + \frac{1}{10} a^{6} - \frac{3}{10} a^{5} + \frac{2}{5} a^{4} - \frac{1}{10} a^{3} - \frac{1}{10} a^{2}$, $\frac{1}{10} a^{14} + \frac{1}{10} a^{10} - \frac{1}{10} a^{9} + \frac{3}{10} a^{8} - \frac{2}{5} a^{7} - \frac{3}{10} a^{6} + \frac{1}{10} a^{5} - \frac{2}{5} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{10} a^{15} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{3}{10} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{10} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{1150} a^{16} + \frac{1}{115} a^{15} - \frac{43}{1150} a^{14} - \frac{22}{575} a^{13} - \frac{51}{1150} a^{12} - \frac{21}{575} a^{11} - \frac{13}{1150} a^{10} - \frac{27}{575} a^{9} + \frac{239}{575} a^{8} + \frac{491}{1150} a^{7} + \frac{1}{5} a^{6} - \frac{163}{575} a^{5} - \frac{53}{575} a^{4} + \frac{413}{1150} a^{3} - \frac{252}{575} a^{2} - \frac{93}{230} a + \frac{15}{46}$, $\frac{1}{353050} a^{17} - \frac{61}{176525} a^{16} + \frac{641}{176525} a^{15} + \frac{2356}{176525} a^{14} + \frac{8057}{353050} a^{13} - \frac{294}{7061} a^{12} - \frac{857}{176525} a^{11} + \frac{16356}{176525} a^{10} + \frac{41178}{176525} a^{9} + \frac{10748}{35305} a^{8} - \frac{163137}{353050} a^{7} - \frac{84851}{353050} a^{6} + \frac{66041}{353050} a^{5} + \frac{14171}{35305} a^{4} + \frac{501}{35305} a^{3} - \frac{81827}{353050} a^{2} - \frac{317}{1535} a + \frac{6185}{14122}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1689}{7675} a^{17} + \frac{334}{307} a^{16} - \frac{61761}{15350} a^{15} + \frac{91586}{7675} a^{14} - \frac{474697}{15350} a^{13} + \frac{1007231}{15350} a^{12} - \frac{1826471}{15350} a^{11} + \frac{1413036}{7675} a^{10} - \frac{1923702}{7675} a^{9} + \frac{4503967}{15350} a^{8} - \frac{183397}{614} a^{7} + \frac{3921123}{15350} a^{6} - \frac{1464101}{7675} a^{5} + \frac{940403}{7675} a^{4} - \frac{546509}{7675} a^{3} + \frac{47448}{1535} a^{2} - \frac{15427}{1535} a + \frac{438}{307} \) (order $6$)
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:  \( 8731.342696996677 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 6.0.34992.1, 9.3.18443443392.1

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

Sibling fields

Degree 9 siblings: data not computed
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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
3Data not computed
$11$11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$