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Properties

Label 18.18.5246823757...4768.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{27}\cdot 3^{24}\cdot 7^{12}$
Root discriminant $44.78$
Ramified primes $2, 3, 7$
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![-3527, 2700, 26814, -14864, -75591, 27684, 104332, -22602, -76548, 9044, 30984, -1866, -6966, 192, 837, -8, -48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 48*x^16 - 8*x^15 + 837*x^14 + 192*x^13 - 6966*x^12 - 1866*x^11 + 30984*x^10 + 9044*x^9 - 76548*x^8 - 22602*x^7 + 104332*x^6 + 27684*x^5 - 75591*x^4 - 14864*x^3 + 26814*x^2 + 2700*x - 3527)
 
gp: K = bnfinit(x^18 - 48*x^16 - 8*x^15 + 837*x^14 + 192*x^13 - 6966*x^12 - 1866*x^11 + 30984*x^10 + 9044*x^9 - 76548*x^8 - 22602*x^7 + 104332*x^6 + 27684*x^5 - 75591*x^4 - 14864*x^3 + 26814*x^2 + 2700*x - 3527, 1)
 

Normalized defining polynomial

\( x^{18} - 48 x^{16} - 8 x^{15} + 837 x^{14} + 192 x^{13} - 6966 x^{12} - 1866 x^{11} + 30984 x^{10} + 9044 x^{9} - 76548 x^{8} - 22602 x^{7} + 104332 x^{6} + 27684 x^{5} - 75591 x^{4} - 14864 x^{3} + 26814 x^{2} + 2700 x - 3527 \)

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:  \(524682375772545974113841184768=2^{27}\cdot 3^{24}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.78$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(504=2^{3}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{504}(1,·)$, $\chi_{504}(193,·)$, $\chi_{504}(457,·)$, $\chi_{504}(205,·)$, $\chi_{504}(337,·)$, $\chi_{504}(277,·)$, $\chi_{504}(25,·)$, $\chi_{504}(37,·)$, $\chi_{504}(289,·)$, $\chi_{504}(421,·)$, $\chi_{504}(169,·)$, $\chi_{504}(109,·)$, $\chi_{504}(445,·)$, $\chi_{504}(373,·)$, $\chi_{504}(361,·)$, $\chi_{504}(121,·)$, $\chi_{504}(253,·)$, $\chi_{504}(85,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{7} + \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{8} + \frac{1}{4} a$, $\frac{1}{2032} a^{16} - \frac{69}{1016} a^{15} + \frac{13}{2032} a^{14} + \frac{159}{508} a^{13} - \frac{231}{1016} a^{12} - \frac{15}{254} a^{11} - \frac{133}{508} a^{10} + \frac{259}{1016} a^{9} + \frac{7}{254} a^{8} + \frac{485}{1016} a^{7} + \frac{38}{127} a^{6} - \frac{23}{127} a^{5} + \frac{195}{508} a^{4} - \frac{245}{508} a^{3} - \frac{867}{2032} a^{2} - \frac{251}{1016} a - \frac{353}{2032}$, $\frac{1}{41145483054823023938747632} a^{17} + \frac{247365765127271087268}{2571592690926438996171727} a^{16} - \frac{1258726957549265437257359}{41145483054823023938747632} a^{15} - \frac{2079588196904441074178955}{20572741527411511969373816} a^{14} + \frac{4953235426010305690515917}{20572741527411511969373816} a^{13} + \frac{4764532846095394740801605}{10286370763705755984686908} a^{12} - \frac{1478119920411530747687133}{10286370763705755984686908} a^{11} + \frac{9168635601328843193364159}{20572741527411511969373816} a^{10} - \frac{1265735475250236457594899}{10286370763705755984686908} a^{9} + \frac{1372726514884865314794445}{20572741527411511969373816} a^{8} - \frac{3112237955562757349375705}{10286370763705755984686908} a^{7} + \frac{864319186937130727506188}{2571592690926438996171727} a^{6} + \frac{446182597450028967850783}{10286370763705755984686908} a^{5} - \frac{944681278569248523596311}{10286370763705755984686908} a^{4} - \frac{18951033074667130430214251}{41145483054823023938747632} a^{3} + \frac{496185060293696464205815}{10286370763705755984686908} a^{2} - \frac{56375217672307116664965}{41145483054823023938747632} a + \frac{8965210133965361976509369}{20572741527411511969373816}$

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:  \( 1222675333.26 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{7})^+\), 6.6.3359232.1, 6.6.8065516032.1, 6.6.8065516032.2, 6.6.1229312.1, 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$