↑

Properties

Label 18.0.31252969564...8983.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{15}\cdot 37^{12}$
Root discriminant $56.20$
Ramified primes $7, 37$
Class number $567$ (GRH)
Class group $[9, 63]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1771561, 1932612, 1947253, 1787533, 1597321, 1403006, 1222838, 417968, 217252, 87838, 38076, 13802, 3610, -778, 203, -36, 13, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 13*x^16 - 36*x^15 + 203*x^14 - 778*x^13 + 3610*x^12 + 13802*x^11 + 38076*x^10 + 87838*x^9 + 217252*x^8 + 417968*x^7 + 1222838*x^6 + 1403006*x^5 + 1597321*x^4 + 1787533*x^3 + 1947253*x^2 + 1932612*x + 1771561)
 
gp: K = bnfinit(x^18 - x^17 + 13*x^16 - 36*x^15 + 203*x^14 - 778*x^13 + 3610*x^12 + 13802*x^11 + 38076*x^10 + 87838*x^9 + 217252*x^8 + 417968*x^7 + 1222838*x^6 + 1403006*x^5 + 1597321*x^4 + 1787533*x^3 + 1947253*x^2 + 1932612*x + 1771561, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 13 x^{16} - 36 x^{15} + 203 x^{14} - 778 x^{13} + 3610 x^{12} + 13802 x^{11} + 38076 x^{10} + 87838 x^{9} + 217252 x^{8} + 417968 x^{7} + 1222838 x^{6} + 1403006 x^{5} + 1597321 x^{4} + 1787533 x^{3} + 1947253 x^{2} + 1932612 x + 1771561 \)

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:  \(-31252969564728218452784752298983=-\,7^{15}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.20$
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:  $7, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(259=7\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{259}(1,·)$, $\chi_{259}(195,·)$, $\chi_{259}(137,·)$, $\chi_{259}(10,·)$, $\chi_{259}(75,·)$, $\chi_{259}(211,·)$, $\chi_{259}(149,·)$, $\chi_{259}(26,·)$, $\chi_{259}(158,·)$, $\chi_{259}(223,·)$, $\chi_{259}(100,·)$, $\chi_{259}(38,·)$, $\chi_{259}(232,·)$, $\chi_{259}(174,·)$, $\chi_{259}(47,·)$, $\chi_{259}(248,·)$, $\chi_{259}(121,·)$, $\chi_{259}(186,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{15141455582} a^{13} + \frac{3170540823}{15141455582} a^{12} - \frac{104500911}{15141455582} a^{11} + \frac{297351821}{15141455582} a^{10} + \frac{1426327149}{15141455582} a^{9} + \frac{1645702362}{7570727791} a^{8} + \frac{1491461621}{7570727791} a^{7} - \frac{1288033783}{15141455582} a^{6} + \frac{7381413561}{15141455582} a^{5} + \frac{2030235529}{15141455582} a^{4} + \frac{2295637533}{15141455582} a^{3} - \frac{6133265819}{15141455582} a^{2} - \frac{1896565093}{7570727791} a + \frac{152790399}{688247981}$, $\frac{1}{333112022804} a^{14} + \frac{5}{166556011402} a^{13} + \frac{4829236143}{83278005701} a^{12} + \frac{28383910647}{166556011402} a^{11} + \frac{4239751029}{166556011402} a^{10} - \frac{9855017757}{83278005701} a^{9} + \frac{4099023924}{83278005701} a^{8} + \frac{7987919801}{166556011402} a^{7} - \frac{64252999319}{166556011402} a^{6} - \frac{32820940682}{83278005701} a^{5} - \frac{43532164873}{166556011402} a^{4} + \frac{45890586715}{166556011402} a^{3} + \frac{35253062056}{83278005701} a^{2} + \frac{1953484880}{7570727791} a + \frac{897757357}{2752991924}$, $\frac{1}{3664232250844} a^{15} - \frac{1}{3664232250844} a^{14} - \frac{27}{916058062711} a^{13} - \frac{301288072851}{1832116125422} a^{12} - \frac{4060955213}{916058062711} a^{11} + \frac{28448643826}{916058062711} a^{10} + \frac{205817200232}{916058062711} a^{9} - \frac{169759173615}{916058062711} a^{8} + \frac{72970675071}{1832116125422} a^{7} - \frac{4453222776}{916058062711} a^{6} - \frac{614017950347}{1832116125422} a^{5} - \frac{406511077574}{916058062711} a^{4} + \frac{435621076872}{916058062711} a^{3} + \frac{32135443266}{83278005701} a^{2} - \frac{311178851}{2752991924} a + \frac{812151}{64023068}$, $\frac{1}{40306554759284} a^{16} - \frac{1}{40306554759284} a^{15} + \frac{13}{40306554759284} a^{14} - \frac{9}{10076638689821} a^{13} - \frac{1291469912542}{10076638689821} a^{12} - \frac{2885319263897}{20153277379642} a^{11} + \frac{1059978716275}{10076638689821} a^{10} - \frac{851489217489}{10076638689821} a^{9} - \frac{1578852610481}{20153277379642} a^{8} - \frac{1869865608021}{20153277379642} a^{7} - \frac{4210120811407}{10076638689821} a^{6} + \frac{1799407554442}{10076638689821} a^{5} - \frac{8388433488957}{20153277379642} a^{4} - \frac{316032168218}{916058062711} a^{3} - \frac{153531601761}{333112022804} a^{2} - \frac{3041832463}{30282911164} a + \frac{1003583921}{2752991924}$, $\frac{1}{443372102352124} a^{17} - \frac{1}{443372102352124} a^{16} + \frac{13}{443372102352124} a^{15} - \frac{9}{110843025588031} a^{14} - \frac{7219}{221686051176062} a^{13} + \frac{23882669392140}{110843025588031} a^{12} + \frac{53541635723247}{221686051176062} a^{11} - \frac{27944226084221}{221686051176062} a^{10} + \frac{17091051274037}{110843025588031} a^{9} + \frac{19558210888918}{110843025588031} a^{8} + \frac{13397142197297}{221686051176062} a^{7} - \frac{90658691393077}{221686051176062} a^{6} + \frac{45879198129493}{110843025588031} a^{5} - \frac{370718074171}{20153277379642} a^{4} + \frac{990970899073}{3664232250844} a^{3} + \frac{151829297423}{333112022804} a^{2} + \frac{1354157805}{2752991924} a + \frac{3733595}{1376495962}$

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

Class group and class number

$C_{9}\times C_{63}$, which has order $567$ (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:  \( \frac{25939}{333112022804} a^{16} + \frac{4360731}{1936697807} a^{9} - \frac{250668723911}{333112022804} a^{2} \) (order $14$)
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:  \( 5877099.24292 \) (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

\(\Q(\sqrt{-7}) \), 3.3.1369.1, 3.3.67081.2, \(\Q(\zeta_{7})^+\), 3.3.67081.1, 6.0.642837223.1, 6.0.31499023927.1, \(\Q(\zeta_{7})\), 6.0.31499023927.2, 9.9.301855146292441.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
7Data not computed
$37$37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$