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Properties

Label 18.18.7635133454...3125.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{24}\cdot 5^{9}\cdot 7^{12}$
Root discriminant $35.40$
Ramified primes $3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-71, -330, 1902, 2893, -11091, -11088, 24296, 18921, -24138, -14385, 12252, 5340, -3347, -996, 492, 89, -36, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 36*x^16 + 89*x^15 + 492*x^14 - 996*x^13 - 3347*x^12 + 5340*x^11 + 12252*x^10 - 14385*x^9 - 24138*x^8 + 18921*x^7 + 24296*x^6 - 11088*x^5 - 11091*x^4 + 2893*x^3 + 1902*x^2 - 330*x - 71)
 
gp: K = bnfinit(x^18 - 3*x^17 - 36*x^16 + 89*x^15 + 492*x^14 - 996*x^13 - 3347*x^12 + 5340*x^11 + 12252*x^10 - 14385*x^9 - 24138*x^8 + 18921*x^7 + 24296*x^6 - 11088*x^5 - 11091*x^4 + 2893*x^3 + 1902*x^2 - 330*x - 71, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 36 x^{16} + 89 x^{15} + 492 x^{14} - 996 x^{13} - 3347 x^{12} + 5340 x^{11} + 12252 x^{10} - 14385 x^{9} - 24138 x^{8} + 18921 x^{7} + 24296 x^{6} - 11088 x^{5} - 11091 x^{4} + 2893 x^{3} + 1902 x^{2} - 330 x - 71 \)

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:  \(7635133454060210702501953125=3^{24}\cdot 5^{9}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.40$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{315}(256,·)$, $\chi_{315}(1,·)$, $\chi_{315}(4,·)$, $\chi_{315}(64,·)$, $\chi_{315}(79,·)$, $\chi_{315}(16,·)$, $\chi_{315}(274,·)$, $\chi_{315}(211,·)$, $\chi_{315}(214,·)$, $\chi_{315}(151,·)$, $\chi_{315}(289,·)$, $\chi_{315}(226,·)$, $\chi_{315}(169,·)$, $\chi_{315}(106,·)$, $\chi_{315}(109,·)$, $\chi_{315}(46,·)$, $\chi_{315}(184,·)$, $\chi_{315}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{8661869281452493060292969} a^{17} + \frac{2094404652110616398888454}{8661869281452493060292969} a^{16} - \frac{1397612008892563141744327}{8661869281452493060292969} a^{15} + \frac{1636648657117081041392967}{8661869281452493060292969} a^{14} - \frac{2847167203377133306782303}{8661869281452493060292969} a^{13} - \frac{371549762614779750861692}{8661869281452493060292969} a^{12} + \frac{3961107087699248250006821}{8661869281452493060292969} a^{11} - \frac{3472364579334646654081569}{8661869281452493060292969} a^{10} + \frac{121278441415286051732587}{8661869281452493060292969} a^{9} + \frac{1072768903331879785057363}{8661869281452493060292969} a^{8} - \frac{2678127001451225468453418}{8661869281452493060292969} a^{7} + \frac{3634376237615523079701155}{8661869281452493060292969} a^{6} - \frac{863782656754774709460207}{8661869281452493060292969} a^{5} - \frac{2377371334409451905177834}{8661869281452493060292969} a^{4} - \frac{2331501751098984219040488}{8661869281452493060292969} a^{3} + \frac{1627723335719082498124771}{8661869281452493060292969} a^{2} + \frac{2046683723170467449331803}{8661869281452493060292969} a + \frac{8358905949799911723124}{121998158893697085356239}$

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:  \( 103946990.673 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{7})^+\), 6.6.820125.1, 6.6.1969120125.1, 6.6.1969120125.2, 6.6.300125.1, 9.9.62523502209.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed