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Properties

Label 18.0.16510744955...0000.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 5^{9}\cdot 7^{12}\cdot 13^{12}$
Root discriminant $90.48$
Ramified primes $2, 5, 7, 13$
Class number $80352$ (GRH)
Class group $[2, 2, 2, 18, 558]$ (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![219558249, 35149140, 129463023, 21380228, 37070685, 5813816, 6632797, 822724, 817621, 72264, 75533, 4488, 5100, 172, 306, -16, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 5*x^16 - 16*x^15 + 306*x^14 + 172*x^13 + 5100*x^12 + 4488*x^11 + 75533*x^10 + 72264*x^9 + 817621*x^8 + 822724*x^7 + 6632797*x^6 + 5813816*x^5 + 37070685*x^4 + 21380228*x^3 + 129463023*x^2 + 35149140*x + 219558249)
 
gp: K = bnfinit(x^18 + 5*x^16 - 16*x^15 + 306*x^14 + 172*x^13 + 5100*x^12 + 4488*x^11 + 75533*x^10 + 72264*x^9 + 817621*x^8 + 822724*x^7 + 6632797*x^6 + 5813816*x^5 + 37070685*x^4 + 21380228*x^3 + 129463023*x^2 + 35149140*x + 219558249, 1)
 

Normalized defining polynomial

\( x^{18} + 5 x^{16} - 16 x^{15} + 306 x^{14} + 172 x^{13} + 5100 x^{12} + 4488 x^{11} + 75533 x^{10} + 72264 x^{9} + 817621 x^{8} + 822724 x^{7} + 6632797 x^{6} + 5813816 x^{5} + 37070685 x^{4} + 21380228 x^{3} + 129463023 x^{2} + 35149140 x + 219558249 \)

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:  \(-165107449555765648724828672000000000=-\,2^{18}\cdot 5^{9}\cdot 7^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.48$
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:  $2, 5, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1820=2^{2}\cdot 5\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1820}(1,·)$, $\chi_{1820}(261,·)$, $\chi_{1820}(841,·)$, $\chi_{1820}(919,·)$, $\chi_{1820}(781,·)$, $\chi_{1820}(1101,·)$, $\chi_{1820}(81,·)$, $\chi_{1820}(659,·)$, $\chi_{1820}(1621,·)$, $\chi_{1820}(599,·)$, $\chi_{1820}(79,·)$, $\chi_{1820}(1439,·)$, $\chi_{1820}(1121,·)$, $\chi_{1820}(1381,·)$, $\chi_{1820}(1639,·)$, $\chi_{1820}(939,·)$, $\chi_{1820}(1199,·)$, $\chi_{1820}(1719,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{12} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{12} a^{4} + \frac{1}{6} a^{3} + \frac{5}{12} a^{2} + \frac{1}{3} a - \frac{1}{4}$, $\frac{1}{12} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{12} a^{3} + \frac{1}{12} a$, $\frac{1}{12} a^{10} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{5}{12} a^{4} - \frac{5}{12} a^{2} - \frac{1}{2}$, $\frac{1}{12} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{5}{12} a^{5} - \frac{1}{2} a^{4} - \frac{5}{12} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{12} a^{12} - \frac{1}{2} a^{7} + \frac{1}{12} a^{6} - \frac{1}{6} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{12} a^{13} + \frac{1}{12} a^{7} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{12} a^{14} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a + \frac{1}{4}$, $\frac{1}{3853103052} a^{15} + \frac{9485651}{1926551526} a^{14} - \frac{58384327}{1926551526} a^{13} - \frac{20901242}{963275763} a^{12} + \frac{44598151}{1926551526} a^{11} - \frac{24801833}{642183842} a^{10} - \frac{40772723}{3853103052} a^{9} - \frac{2009444}{321091921} a^{8} - \frac{141460295}{642183842} a^{7} - \frac{202060826}{963275763} a^{6} + \frac{346280347}{3853103052} a^{5} + \frac{98746257}{642183842} a^{4} - \frac{175014343}{1284367684} a^{3} + \frac{257431006}{963275763} a^{2} + \frac{828490411}{1926551526} a + \frac{197069623}{642183842}$, $\frac{1}{8368939828944} a^{16} - \frac{83}{2092234957236} a^{15} - \frac{4721234855}{348705826206} a^{14} - \frac{84303838237}{2092234957236} a^{13} - \frac{97865365667}{4184469914472} a^{12} + \frac{3432597555}{232470550804} a^{11} + \frac{16943813551}{1394823304824} a^{10} + \frac{2459450923}{348705826206} a^{9} - \frac{170765656357}{8368939828944} a^{8} - \frac{193427676562}{523058739309} a^{7} - \frac{607671601787}{1394823304824} a^{6} - \frac{481626682073}{2092234957236} a^{5} - \frac{1456992216649}{8368939828944} a^{4} + \frac{224488238785}{523058739309} a^{3} - \frac{511714521823}{4184469914472} a^{2} + \frac{58584277214}{174352913103} a - \frac{1975932423}{5137470736}$, $\frac{1}{1642844539553204349552238241348304} a^{17} + \frac{25584735866829022315}{1642844539553204349552238241348304} a^{16} - \frac{2503725143478693473453}{34225927907358423949004963361423} a^{15} - \frac{9414962096782653723016683956899}{410711134888301087388059560337076} a^{14} - \frac{23227032551685001461964515622445}{821422269776602174776119120674152} a^{13} + \frac{3090506737711733909688277997585}{91269141086289130530679902297128} a^{12} + \frac{1033789580213089338997139523729}{91269141086289130530679902297128} a^{11} - \frac{16341405218249892372934179943}{91269141086289130530679902297128} a^{10} - \frac{50778668323178962528386708401221}{1642844539553204349552238241348304} a^{9} + \frac{8840576475807250470413538549497}{1642844539553204349552238241348304} a^{8} + \frac{26167041589889743126041467105195}{273807423258867391592039706891384} a^{7} + \frac{237910511573628348670232813127041}{821422269776602174776119120674152} a^{6} - \frac{564597473062291828633609947312289}{1642844539553204349552238241348304} a^{5} + \frac{376480430676351771106429877894857}{1642844539553204349552238241348304} a^{4} - \frac{386087751498598217324927140641985}{821422269776602174776119120674152} a^{3} + \frac{13460530662748127727069325067449}{273807423258867391592039706891384} a^{2} - \frac{87683728273390529688455766206327}{182538282172578261061359804594256} a + \frac{176485280678601632069916613851}{1008498796533581552825192290576}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{18}\times C_{558}$, which has order $80352$ (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:  \( 205236.825908 \) (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

\(\Q(\sqrt{-5}) \), 3.3.8281.1, \(\Q(\zeta_{7})^+\), 3.3.8281.2, 3.3.169.1, 6.0.548599688000.3, 6.0.19208000.1, 6.0.548599688000.2, 6.0.228488000.1, 9.9.567869252041.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$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}$
$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}$
$13$13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$