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Properties

Label 18.0.34699574084...4375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 5^{9}\cdot 13^{12}$
Root discriminant $64.24$
Ramified primes $3, 5, 13$
Class number $2744$ (GRH)
Class group $[2, 14, 98]$ (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![1307431, 711594, 602835, 147158, 123351, -52260, 6863, -27264, 26967, 6720, 3765, 372, -486, -84, 48, 10, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 9*x^16 + 10*x^15 + 48*x^14 - 84*x^13 - 486*x^12 + 372*x^11 + 3765*x^10 + 6720*x^9 + 26967*x^8 - 27264*x^7 + 6863*x^6 - 52260*x^5 + 123351*x^4 + 147158*x^3 + 602835*x^2 + 711594*x + 1307431)
 
gp: K = bnfinit(x^18 - 6*x^17 + 9*x^16 + 10*x^15 + 48*x^14 - 84*x^13 - 486*x^12 + 372*x^11 + 3765*x^10 + 6720*x^9 + 26967*x^8 - 27264*x^7 + 6863*x^6 - 52260*x^5 + 123351*x^4 + 147158*x^3 + 602835*x^2 + 711594*x + 1307431, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 9 x^{16} + 10 x^{15} + 48 x^{14} - 84 x^{13} - 486 x^{12} + 372 x^{11} + 3765 x^{10} + 6720 x^{9} + 26967 x^{8} - 27264 x^{7} + 6863 x^{6} - 52260 x^{5} + 123351 x^{4} + 147158 x^{3} + 602835 x^{2} + 711594 x + 1307431 \)

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:  \(-346995740849617491120493646484375=-\,3^{27}\cdot 5^{9}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.24$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(585=3^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{585}(256,·)$, $\chi_{585}(1,·)$, $\chi_{585}(451,·)$, $\chi_{585}(196,·)$, $\chi_{585}(391,·)$, $\chi_{585}(74,·)$, $\chi_{585}(269,·)$, $\chi_{585}(14,·)$, $\chi_{585}(464,·)$, $\chi_{585}(209,·)$, $\chi_{585}(211,·)$, $\chi_{585}(404,·)$, $\chi_{585}(406,·)$, $\chi_{585}(29,·)$, $\chi_{585}(224,·)$, $\chi_{585}(16,·)$, $\chi_{585}(419,·)$, $\chi_{585}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{12} + \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{3}{16} a^{5} + \frac{3}{16} a^{4} - \frac{1}{16} a^{3} - \frac{1}{2} a^{2} + \frac{3}{16} a - \frac{3}{16}$, $\frac{1}{16} a^{13} + \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{3}{16} a^{6} + \frac{3}{16} a^{5} + \frac{3}{16} a^{4} - \frac{1}{16} a^{2} + \frac{5}{16} a + \frac{1}{4}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{11} + \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{3}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{3}{16} a^{4} - \frac{3}{16} a^{2} + \frac{1}{16} a + \frac{7}{16}$, $\frac{1}{272} a^{15} + \frac{7}{272} a^{13} + \frac{1}{68} a^{12} + \frac{3}{34} a^{11} + \frac{5}{272} a^{10} - \frac{33}{272} a^{9} + \frac{15}{272} a^{8} + \frac{9}{272} a^{7} + \frac{49}{272} a^{6} - \frac{65}{272} a^{5} - \frac{2}{17} a^{4} - \frac{31}{68} a^{3} - \frac{5}{136} a^{2} + \frac{125}{272} a - \frac{55}{272}$, $\frac{1}{28832} a^{16} + \frac{15}{14416} a^{15} - \frac{41}{1802} a^{14} + \frac{447}{14416} a^{13} + \frac{123}{14416} a^{12} - \frac{73}{3604} a^{11} + \frac{195}{7208} a^{10} - \frac{139}{14416} a^{9} - \frac{59}{1696} a^{8} + \frac{2293}{14416} a^{7} - \frac{197}{7208} a^{6} - \frac{2691}{14416} a^{5} - \frac{7085}{28832} a^{4} - \frac{1281}{7208} a^{3} + \frac{6551}{14416} a^{2} - \frac{6219}{14416} a + \frac{9043}{28832}$, $\frac{1}{14179287874115390838017359700639195171776} a^{17} + \frac{178218420864885686392686625142572133}{14179287874115390838017359700639195171776} a^{16} - \frac{267336301106127297035209361104631679}{886205492132211927376084981289949698236} a^{15} + \frac{103340196497870272036551679559645944381}{7089643937057695419008679850319597585888} a^{14} - \frac{31478259623672220765373434810077716419}{7089643937057695419008679850319597585888} a^{13} - \frac{13502022437165008510899224260477925989}{7089643937057695419008679850319597585888} a^{12} + \frac{17327414054034855296466019413723532063}{886205492132211927376084981289949698236} a^{11} - \frac{6582615268496427271369532882777684223}{3544821968528847709504339925159798792944} a^{10} + \frac{53344538887566303489371194096583259077}{834075757300905343412785864743482068928} a^{9} + \frac{418835699687269265721669075996116928607}{14179287874115390838017359700639195171776} a^{8} - \frac{780674991516668754609368966217089910067}{3544821968528847709504339925159798792944} a^{7} - \frac{368049544644595745541812097988430404027}{1772410984264423854752169962579899396472} a^{6} - \frac{3252421361942912285277183722520828681977}{14179287874115390838017359700639195171776} a^{5} + \frac{337387504476367947204554219454652528649}{14179287874115390838017359700639195171776} a^{4} - \frac{2116735948673680512932786759579963021365}{7089643937057695419008679850319597585888} a^{3} + \frac{274495354033739763631594570458873427225}{3544821968528847709504339925159798792944} a^{2} + \frac{7067409484308579788633284077795883023087}{14179287874115390838017359700639195171776} a + \frac{7081172259692557125899054151428522561859}{14179287874115390838017359700639195171776}$

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

Class group and class number

$C_{2}\times C_{14}\times C_{98}$, which has order $2744$ (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:  \( 400417.136445 \) (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

\(\Q(\sqrt{-15}) \), \(\Q(\zeta_{9})^+\), 3.3.169.1, 3.3.13689.1, 3.3.13689.2, 6.0.2460375.1, 6.0.96393375.1, 6.0.70270770375.5, 6.0.70270770375.7, 9.9.2565164201769.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13Data not computed