↑

Properties

Label 18.10.1159892457...8517.1
Degree $18$
Signature $[10, 4]$
Discriminant $3^{24}\cdot 7^{12}\cdot 197^{5}$
Root discriminant $68.69$
Ramified primes $3, 7, 197$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T459

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1227113, 5503983, -6404370, -9132658, 3882756, 4986567, 1096380, -651636, -380205, -47780, -34395, 20355, 25, 1938, 51, -108, 24, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 24*x^16 - 108*x^15 + 51*x^14 + 1938*x^13 + 25*x^12 + 20355*x^11 - 34395*x^10 - 47780*x^9 - 380205*x^8 - 651636*x^7 + 1096380*x^6 + 4986567*x^5 + 3882756*x^4 - 9132658*x^3 - 6404370*x^2 + 5503983*x + 1227113)
 
gp: K = bnfinit(x^18 - 9*x^17 + 24*x^16 - 108*x^15 + 51*x^14 + 1938*x^13 + 25*x^12 + 20355*x^11 - 34395*x^10 - 47780*x^9 - 380205*x^8 - 651636*x^7 + 1096380*x^6 + 4986567*x^5 + 3882756*x^4 - 9132658*x^3 - 6404370*x^2 + 5503983*x + 1227113, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 24 x^{16} - 108 x^{15} + 51 x^{14} + 1938 x^{13} + 25 x^{12} + 20355 x^{11} - 34395 x^{10} - 47780 x^{9} - 380205 x^{8} - 651636 x^{7} + 1096380 x^{6} + 4986567 x^{5} + 3882756 x^{4} - 9132658 x^{3} - 6404370 x^{2} + 5503983 x + 1227113 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1159892457286612080082548844598517=3^{24}\cdot 7^{12}\cdot 197^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.69$
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:  $3, 7, 197$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{16} + \frac{1}{8} a^{15} + \frac{1}{8} a^{14} + \frac{1}{8} a^{13} - \frac{3}{8} a^{11} + \frac{3}{8} a^{10} - \frac{1}{4} a^{8} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{3}{8} a^{4} - \frac{1}{8} a^{3} + \frac{3}{8} a^{2} + \frac{1}{8} a - \frac{3}{8}$, $\frac{1}{22070108234025388192497636415069999413967644800176934289777989696} a^{17} + \frac{183566903850520970855462218954611977449076942512520350098714403}{5517527058506347048124409103767499853491911200044233572444497424} a^{16} - \frac{842445235516609422106191742886030803734887886917164391837190715}{5517527058506347048124409103767499853491911200044233572444497424} a^{15} + \frac{69127060127208342749389085487476097349267626757249267234160895}{2758763529253173524062204551883749926745955600022116786222248712} a^{14} + \frac{1309732047859709061155251344136874880365317596872115783117502283}{22070108234025388192497636415069999413967644800176934289777989696} a^{13} - \frac{10124867264468437683607410609104341200099058801416941816776158663}{22070108234025388192497636415069999413967644800176934289777989696} a^{12} - \frac{1593184970425206045858737209046071003870877175813900433032609757}{11035054117012694096248818207534999706983822400088467144888994848} a^{11} + \frac{1071305308900305887414377768395838479961460391534865432508948673}{22070108234025388192497636415069999413967644800176934289777989696} a^{10} - \frac{901722997156109552678571204973512589627362980841207355315288931}{11035054117012694096248818207534999706983822400088467144888994848} a^{9} + \frac{4752657686703360601347817189163202843556101274271758676965580079}{11035054117012694096248818207534999706983822400088467144888994848} a^{8} - \frac{10880590856414175591082888884901339027998482288248566938600645943}{22070108234025388192497636415069999413967644800176934289777989696} a^{7} + \frac{4393122051451435434021787239231980329796487725282129625702719369}{22070108234025388192497636415069999413967644800176934289777989696} a^{6} - \frac{3085305622216754793215188300008753768748059316399603857015334151}{22070108234025388192497636415069999413967644800176934289777989696} a^{5} + \frac{966293959576881976677386919244193395356936915396311505934703741}{5517527058506347048124409103767499853491911200044233572444497424} a^{4} + \frac{1016373705896271285288420955329011233759828840999130289987864785}{2758763529253173524062204551883749926745955600022116786222248712} a^{3} + \frac{3146005152466101931600417468774995189409295146330925334925280347}{11035054117012694096248818207534999706983822400088467144888994848} a^{2} - \frac{2189741837048147691486674831588111749552010068173751203809523177}{5517527058506347048124409103767499853491911200044233572444497424} a - \frac{2006764054302696669727070088858617668646303096438230692238840645}{22070108234025388192497636415069999413967644800176934289777989696}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $13$
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:  \( 5488097532.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T459:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4608
The 96 conjugacy class representatives for t18n459 are not computed
Character table for t18n459 is not computed

Intermediate fields

3.3.3969.2, \(\Q(\zeta_{9})^+\), 3.3.3969.1, \(\Q(\zeta_{7})^+\), 9.9.62523502209.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

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 ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$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}$
197Data not computed