↑

Properties

Label 18.0.97485335078...5519.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{45}\cdot 53^{9}$
Root discriminant $113.49$
Ramified primes $3, 53$
Class number $2076160$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 8110]$ (GRH)
Galois group $C_{18}$ (as 18T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![139161467719, -84219534360, 66074188401, -21594752400, 33884199180, -1495021320, 6689957274, -38333880, 661644126, -327640, 36758007, 0, 1199562, 0, 22815, 0, 234, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 234*x^16 + 22815*x^14 + 1199562*x^12 + 36758007*x^10 - 327640*x^9 + 661644126*x^8 - 38333880*x^7 + 6689957274*x^6 - 1495021320*x^5 + 33884199180*x^4 - 21594752400*x^3 + 66074188401*x^2 - 84219534360*x + 139161467719)
 
gp: K = bnfinit(x^18 + 234*x^16 + 22815*x^14 + 1199562*x^12 + 36758007*x^10 - 327640*x^9 + 661644126*x^8 - 38333880*x^7 + 6689957274*x^6 - 1495021320*x^5 + 33884199180*x^4 - 21594752400*x^3 + 66074188401*x^2 - 84219534360*x + 139161467719, 1)
 

Normalized defining polynomial

\( x^{18} + 234 x^{16} + 22815 x^{14} + 1199562 x^{12} + 36758007 x^{10} - 327640 x^{9} + 661644126 x^{8} - 38333880 x^{7} + 6689957274 x^{6} - 1495021320 x^{5} + 33884199180 x^{4} - 21594752400 x^{3} + 66074188401 x^{2} - 84219534360 x + 139161467719 \)

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:  \(-9748533507874859943721777394206605519=-\,3^{45}\cdot 53^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $113.49$
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:  $3, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1431=3^{3}\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{1431}(1,·)$, $\chi_{1431}(1430,·)$, $\chi_{1431}(953,·)$, $\chi_{1431}(1112,·)$, $\chi_{1431}(1114,·)$, $\chi_{1431}(476,·)$, $\chi_{1431}(794,·)$, $\chi_{1431}(158,·)$, $\chi_{1431}(160,·)$, $\chi_{1431}(635,·)$, $\chi_{1431}(796,·)$, $\chi_{1431}(637,·)$, $\chi_{1431}(478,·)$, $\chi_{1431}(1271,·)$, $\chi_{1431}(1273,·)$, $\chi_{1431}(955,·)$, $\chi_{1431}(317,·)$, $\chi_{1431}(319,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{53158} a^{9} + \frac{117}{53158} a^{7} + \frac{4563}{53158} a^{5} + \frac{6376}{26579} a^{3} - \frac{8741}{53158} a + \frac{22233}{53158}$, $\frac{1}{53158} a^{10} + \frac{117}{53158} a^{8} + \frac{4563}{53158} a^{6} + \frac{6376}{26579} a^{4} - \frac{8741}{53158} a^{2} + \frac{22233}{53158} a$, $\frac{1}{53158} a^{11} - \frac{4563}{26579} a^{7} + \frac{10461}{53158} a^{5} - \frac{12301}{53158} a^{3} + \frac{22233}{53158} a^{2} + \frac{12695}{53158} a + \frac{3481}{53158}$, $\frac{1}{53158} a^{12} - \frac{4563}{26579} a^{8} + \frac{10461}{53158} a^{6} - \frac{12301}{53158} a^{4} + \frac{22233}{53158} a^{3} + \frac{12695}{53158} a^{2} + \frac{3481}{53158} a$, $\frac{1}{53158} a^{13} + \frac{2149}{7594} a^{7} + \frac{989}{7594} a^{5} + \frac{22233}{53158} a^{4} + \frac{24585}{53158} a^{3} + \frac{3481}{53158} a^{2} + \frac{9896}{26579} a - \frac{2864}{26579}$, $\frac{1}{6270117345666734} a^{14} + \frac{18604946345}{3135058672833367} a^{13} + \frac{13}{447865524690481} a^{12} + \frac{5284429863}{895731049380962} a^{11} + \frac{1859}{895731049380962} a^{10} + \frac{45366208205}{6270117345666734} a^{9} + \frac{32955}{447865524690481} a^{8} - \frac{3094954837314227}{6270117345666734} a^{7} + \frac{599781}{447865524690481} a^{6} + \frac{2476647931395601}{6270117345666734} a^{5} - \frac{446450022289377}{6270117345666734} a^{4} + \frac{69237905488091}{3135058672833367} a^{3} + \frac{409236209358644}{3135058672833367} a^{2} + \frac{2892311771861853}{6270117345666734} a + \frac{111195486500192}{447865524690481}$, $\frac{1}{6270117345666734} a^{15} + \frac{195}{6270117345666734} a^{13} - \frac{5959364539}{3135058672833367} a^{12} + \frac{7605}{3135058672833367} a^{11} + \frac{13958883700}{3135058672833367} a^{10} + \frac{604175}{6270117345666734} a^{9} + \frac{56018350175814}{3135058672833367} a^{8} + \frac{6426225}{3135058672833367} a^{7} + \frac{924393536037081}{6270117345666734} a^{6} + \frac{10024911}{447865524690481} a^{5} - \frac{18046504138569}{3135058672833367} a^{4} - \frac{2516751154723641}{6270117345666734} a^{3} - \frac{2105803435213209}{6270117345666734} a^{2} + \frac{16909396175469}{447865524690481} a - \frac{302311458152719}{3135058672833367}$, $\frac{1}{6270117345666734} a^{16} + \frac{22602636349}{3135058672833367} a^{13} - \frac{10140}{3135058672833367} a^{12} + \frac{4885805879}{3135058672833367} a^{11} - \frac{966680}{3135058672833367} a^{10} - \frac{9060299861}{3135058672833367} a^{9} - \frac{38557350}{3135058672833367} a^{8} - \frac{2838997172337857}{6270117345666734} a^{7} - \frac{106932384}{447865524690481} a^{6} - \frac{52958933544277}{895731049380962} a^{5} + \frac{185538268332729}{447865524690481} a^{4} + \frac{42855533734779}{3135058672833367} a^{3} + \frac{979550075888525}{6270117345666734} a^{2} + \frac{3040366693319835}{6270117345666734} a - \frac{200177575807996}{3135058672833367}$, $\frac{1}{6270117345666734} a^{17} - \frac{11492}{3135058672833367} a^{13} + \frac{19542404416}{3135058672833367} a^{12} - \frac{1195168}{3135058672833367} a^{11} - \frac{45371450445}{6270117345666734} a^{10} - \frac{53409070}{3135058672833367} a^{9} + \frac{563812652942380}{3135058672833367} a^{8} - \frac{1211900352}{3135058672833367} a^{7} + \frac{795349517950399}{3135058672833367} a^{6} - \frac{1969338072}{447865524690481} a^{5} - \frac{800591441401051}{3135058672833367} a^{4} - \frac{3100988997604189}{6270117345666734} a^{3} + \frac{860134040028082}{3135058672833367} a^{2} + \frac{1502166453397503}{6270117345666734} a + \frac{179609839098212}{3135058672833367}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{8110}$, which has order $2076160$ (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:  \( 40934.03294431194 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-159}) \), \(\Q(\zeta_{9})^+\), 6.0.2930345991.1, \(\Q(\zeta_{27})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ R $18$

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
$53$53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$