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Properties

Label 18.0.12822537788...6875.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{43}\cdot 5^{8}$
Root discriminant $28.21$
Ramified primes $3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3303, 8424, 13284, 15660, 11556, 162, -4617, 972, 5346, 960, -1836, -540, 387, 72, -54, -9, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^15 - 54*x^14 + 72*x^13 + 387*x^12 - 540*x^11 - 1836*x^10 + 960*x^9 + 5346*x^8 + 972*x^7 - 4617*x^6 + 162*x^5 + 11556*x^4 + 15660*x^3 + 13284*x^2 + 8424*x + 3303)
 
gp: K = bnfinit(x^18 - 9*x^15 - 54*x^14 + 72*x^13 + 387*x^12 - 540*x^11 - 1836*x^10 + 960*x^9 + 5346*x^8 + 972*x^7 - 4617*x^6 + 162*x^5 + 11556*x^4 + 15660*x^3 + 13284*x^2 + 8424*x + 3303, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{15} - 54 x^{14} + 72 x^{13} + 387 x^{12} - 540 x^{11} - 1836 x^{10} + 960 x^{9} + 5346 x^{8} + 972 x^{7} - 4617 x^{6} + 162 x^{5} + 11556 x^{4} + 15660 x^{3} + 13284 x^{2} + 8424 x + 3303 \)

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:  \(-128225377888491045948046875=-\,3^{43}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.21$
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:  $3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{3} a^{12}$, $\frac{1}{3} a^{13}$, $\frac{1}{9} a^{14} + \frac{1}{3} a^{5}$, $\frac{1}{9} a^{15} + \frac{1}{3} a^{6}$, $\frac{1}{3015} a^{16} + \frac{103}{3015} a^{15} + \frac{161}{3015} a^{14} - \frac{20}{201} a^{13} + \frac{12}{335} a^{12} - \frac{6}{335} a^{11} + \frac{9}{335} a^{10} + \frac{10}{201} a^{9} + \frac{164}{335} a^{8} - \frac{59}{1005} a^{7} - \frac{116}{1005} a^{6} - \frac{202}{1005} a^{5} - \frac{99}{335} a^{4} + \frac{93}{335} a^{3} + \frac{3}{67} a^{2} + \frac{106}{335} a + \frac{59}{335}$, $\frac{1}{12334074502751072514174676896495} a^{17} - \frac{339684609810712781349292828}{4111358167583690838058225632165} a^{16} + \frac{29944665642070778453972383075}{2466814900550214502834935379299} a^{15} + \frac{658584038526423175856804803823}{12334074502751072514174676896495} a^{14} + \frac{187139031775567583939530024437}{1370452722527896946019408544055} a^{13} - \frac{97167102295170233716637265127}{822271633516738167611645126433} a^{12} - \frac{191877362044420835182572236177}{4111358167583690838058225632165} a^{11} - \frac{93880368198047263311917600303}{1370452722527896946019408544055} a^{10} - \frac{64163858534628086780156640086}{1370452722527896946019408544055} a^{9} - \frac{471278451947377708175497597868}{4111358167583690838058225632165} a^{8} - \frac{45926875081543014449127852251}{1370452722527896946019408544055} a^{7} - \frac{251282762032494029651468930270}{822271633516738167611645126433} a^{6} - \frac{1189775026377768343966282381333}{4111358167583690838058225632165} a^{5} + \frac{58090235180146928416915179206}{1370452722527896946019408544055} a^{4} + \frac{22708510338478711607658561864}{1370452722527896946019408544055} a^{3} + \frac{202908702833844313756320572266}{1370452722527896946019408544055} a^{2} + \frac{368542434223947214597681543327}{1370452722527896946019408544055} a - \frac{294505233208457985114694607648}{1370452722527896946019408544055}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{11318495344569020}{36805902353098824601} a^{17} + \frac{21045731337996484}{36805902353098824601} a^{16} - \frac{62758891250893486}{110417707059296473803} a^{15} + \frac{117673835786826818}{36805902353098824601} a^{14} + \frac{413325837809012696}{36805902353098824601} a^{13} - \frac{1763983852337497839}{36805902353098824601} a^{12} - \frac{1861970029668845248}{36805902353098824601} a^{11} + \frac{11965277934703649096}{36805902353098824601} a^{10} + \frac{8922659867354334449}{110417707059296473803} a^{9} - \frac{32782098580665636678}{36805902353098824601} a^{8} - \frac{14397705110028220404}{36805902353098824601} a^{7} + \frac{57863669014594855619}{36805902353098824601} a^{6} - \frac{4824084155602955394}{36805902353098824601} a^{5} - \frac{50262932969501381304}{36805902353098824601} a^{4} - \frac{70257591514868044263}{36805902353098824601} a^{3} - \frac{13277213549750899398}{36805902353098824601} a^{2} + \frac{274656442026671196}{36805902353098824601} a + \frac{30977460612645265657}{36805902353098824601} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1407082.5964197547 \) (assuming GRH)
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

\(\Q(\sqrt{-3}) \), 3.1.243.1 x3, 6.0.177147.2, 9.3.6537720751875.3

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$5$5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$