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Properties

Label 18.0.14219059916...4459.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 7^{15}\cdot 13^{9}$
Root discriminant $78.96$
Ramified primes $3, 7, 13$
Class number $32984$ (GRH)
Class group $[2, 2, 8246]$ (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![337293181, -173351430, 217719351, -102732573, 88661442, -36408891, 21644553, -7306587, 3240978, -886259, 328884, -71760, 24968, -4017, 1377, -145, 51, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 51*x^16 - 145*x^15 + 1377*x^14 - 4017*x^13 + 24968*x^12 - 71760*x^11 + 328884*x^10 - 886259*x^9 + 3240978*x^8 - 7306587*x^7 + 21644553*x^6 - 36408891*x^5 + 88661442*x^4 - 102732573*x^3 + 217719351*x^2 - 173351430*x + 337293181)
 
gp: K = bnfinit(x^18 - 3*x^17 + 51*x^16 - 145*x^15 + 1377*x^14 - 4017*x^13 + 24968*x^12 - 71760*x^11 + 328884*x^10 - 886259*x^9 + 3240978*x^8 - 7306587*x^7 + 21644553*x^6 - 36408891*x^5 + 88661442*x^4 - 102732573*x^3 + 217719351*x^2 - 173351430*x + 337293181, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 51 x^{16} - 145 x^{15} + 1377 x^{14} - 4017 x^{13} + 24968 x^{12} - 71760 x^{11} + 328884 x^{10} - 886259 x^{9} + 3240978 x^{8} - 7306587 x^{7} + 21644553 x^{6} - 36408891 x^{5} + 88661442 x^{4} - 102732573 x^{3} + 217719351 x^{2} - 173351430 x + 337293181 \)

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:  \(-14219059916154378364791264791424459=-\,3^{24}\cdot 7^{15}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.96$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(819=3^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{819}(1,·)$, $\chi_{819}(454,·)$, $\chi_{819}(649,·)$, $\chi_{819}(781,·)$, $\chi_{819}(79,·)$, $\chi_{819}(274,·)$, $\chi_{819}(727,·)$, $\chi_{819}(220,·)$, $\chi_{819}(352,·)$, $\chi_{819}(547,·)$, $\chi_{819}(103,·)$, $\chi_{819}(235,·)$, $\chi_{819}(493,·)$, $\chi_{819}(625,·)$, $\chi_{819}(181,·)$, $\chi_{819}(376,·)$, $\chi_{819}(508,·)$, $\chi_{819}(766,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{97} a^{12} + \frac{19}{97} a^{11} + \frac{15}{97} a^{10} + \frac{34}{97} a^{9} - \frac{2}{97} a^{8} + \frac{23}{97} a^{7} - \frac{27}{97} a^{6} + \frac{26}{97} a^{5} + \frac{16}{97} a^{4} + \frac{16}{97} a^{3} - \frac{2}{97} a^{2} + \frac{3}{97} a + \frac{32}{97}$, $\frac{1}{97} a^{13} + \frac{42}{97} a^{11} + \frac{40}{97} a^{10} + \frac{31}{97} a^{9} - \frac{36}{97} a^{8} + \frac{21}{97} a^{7} - \frac{43}{97} a^{6} + \frac{7}{97} a^{5} + \frac{3}{97} a^{4} - \frac{15}{97} a^{3} + \frac{41}{97} a^{2} - \frac{25}{97} a - \frac{26}{97}$, $\frac{1}{97} a^{14} + \frac{18}{97} a^{11} - \frac{17}{97} a^{10} - \frac{9}{97} a^{9} + \frac{8}{97} a^{8} - \frac{39}{97} a^{7} - \frac{23}{97} a^{6} - \frac{22}{97} a^{5} - \frac{8}{97} a^{4} + \frac{48}{97} a^{3} - \frac{38}{97} a^{2} + \frac{42}{97} a + \frac{14}{97}$, $\frac{1}{97} a^{15} + \frac{29}{97} a^{11} + \frac{12}{97} a^{10} - \frac{22}{97} a^{9} - \frac{3}{97} a^{8} + \frac{48}{97} a^{7} - \frac{21}{97} a^{6} + \frac{9}{97} a^{5} - \frac{46}{97} a^{4} - \frac{35}{97} a^{3} - \frac{19}{97} a^{2} - \frac{40}{97} a + \frac{6}{97}$, $\frac{1}{97} a^{16} + \frac{43}{97} a^{11} + \frac{28}{97} a^{10} - \frac{19}{97} a^{9} + \frac{9}{97} a^{8} - \frac{9}{97} a^{7} + \frac{16}{97} a^{6} - \frac{24}{97} a^{5} - \frac{14}{97} a^{4} + \frac{2}{97} a^{3} + \frac{18}{97} a^{2} + \frac{16}{97} a + \frac{42}{97}$, $\frac{1}{3130892788963172701507875878277895708666331625983941873723} a^{17} - \frac{14277714492780780505173877161047681528258930514285342394}{3130892788963172701507875878277895708666331625983941873723} a^{16} + \frac{2703141394959332422111931062959516064719589442952171973}{3130892788963172701507875878277895708666331625983941873723} a^{15} + \frac{4151269626118213507966464677819226666374182263181000786}{3130892788963172701507875878277895708666331625983941873723} a^{14} + \frac{10119470104499319692416621573358044762087089820244268552}{3130892788963172701507875878277895708666331625983941873723} a^{13} - \frac{13456221353968869006246294206695440337500640599243557889}{3130892788963172701507875878277895708666331625983941873723} a^{12} - \frac{946581834914187022892711621917207169302538845526348186731}{3130892788963172701507875878277895708666331625983941873723} a^{11} + \frac{1361728349770676353243377106001145272647586811492736043936}{3130892788963172701507875878277895708666331625983941873723} a^{10} - \frac{500007090041938670901011998269937893708160830428028902530}{3130892788963172701507875878277895708666331625983941873723} a^{9} - \frac{11451074173584882636651611600623383069343222888227662885}{3130892788963172701507875878277895708666331625983941873723} a^{8} - \frac{65431170169714652866251902768399585893603820421232564888}{3130892788963172701507875878277895708666331625983941873723} a^{7} - \frac{175054433275796135358070029729331321331513123815078463203}{3130892788963172701507875878277895708666331625983941873723} a^{6} - \frac{394261239910992745875027978966260985904947621213804732602}{3130892788963172701507875878277895708666331625983941873723} a^{5} - \frac{130555862801159363486364351008850310914197001252740986269}{3130892788963172701507875878277895708666331625983941873723} a^{4} - \frac{308509183704068406329651451866476538834140854126612976681}{3130892788963172701507875878277895708666331625983941873723} a^{3} + \frac{1114536516730583682416519346910461934121561657443066669412}{3130892788963172701507875878277895708666331625983941873723} a^{2} - \frac{783935829067288807318271382106418318872348569822794472373}{3130892788963172701507875878277895708666331625983941873723} a + \frac{1078273858643719552700747755795354474407324939917582398772}{3130892788963172701507875878277895708666331625983941873723}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{8246}$, which has order $32984$ (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:  \( 54408.4888887 \) (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

\(\Q(\sqrt{-91}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.1, 3.3.3969.2, 6.0.4944179331.10, 6.0.36924979.1, 6.0.242264787219.4, 6.0.242264787219.5, 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
7Data not computed
$13$13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$